VECTORS AND THE GEOMETRY OF SPACE

12 VECTORS AND THE GEOMETRY OF SPACE OSLO

BERLIN LONDON

PARIS

ROME MADRID LISBON

Wind velocity is a vector because it has both magnitude and direction. Pictured are velocity vectors showing the wind pattern over the North Atlantic and Western Europe on February 28, 2007. Larger arrows indicate stronger winds.

In this chapter we introduce vectors and coordinate systems for three-dimensional space. This will be the setting for our study of the calculus of functions of two variables in Chapter 14 because the graph of such a function is a surface in space. In this chapter we will see that vectors provide particularly simple descriptions of lines and planes in space.

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12.1 THREE-DIMENSIONAL COORDINATE SYSTEMS z

To locate a point in a plane, two numbers are necessary. We know that any point in the plane can be represented as an ordered pair 共a, b兲 of real numbers, where a is the x-coordinate and b is the y-coordinate. For this reason, a plane is called two-dimensional. To locate a point in space, three numbers are required. We represent any point in space by an ordered triple 共a, b, c兲 of real numbers. In order to represent points in space, we first choose a fixed point O (the origin) and three directed lines through O that are perpendicular to each other, called the coordinate axes and labeled the x-axis, y-axis, and z-axis. Usually we think of the x- and y-axes as being horizontal and the z-axis as being vertical, and we draw the orientation of the axes as in Figure 1. The direction of the z-axis is determined by the right-hand rule as illustrated in Figure 2: If you curl the fingers of your right hand around the z-axis in the direction of a 90⬚ counterclockwise rotation from the positive x-axis to the positive y-axis, then your thumb points in the positive direction of the z-axis. The three coordinate axes determine the three coordinate planes illustrated in Figure 3(a). The xy-plane is the plane that contains the x- and y-axes; the yz-plane contains the y- and z-axes; the xz-plane contains the x- and z-axes. These three coordinate planes divide space into eight parts, called octants. The first octant, in the foreground, is determined by the positive axes.

O y x

FIGURE 1

Coordinate axes z

y x

z

FIGURE 2

z

Right-hand rule y z-plan

lane

xz-p

x

FIGURE 3

z P(a, b, c)

a

O

c y

x

FIGURE 4

b

e

left

O

xy-plane (a) Coordinate planes

y

x

right w

l wal O

floor

all y

(b)

Because many people have some difficulty visualizing diagrams of three-dimensional figures, you may find it helpful to do the following [see Figure 3(b)]. Look at any bottom corner of a room and call the corner the origin. The wall on your left is in the xz-plane, the wall on your right is in the yz-plane, and the floor is in the xy-plane. The x-axis runs along the intersection of the floor and the left wall. The y-axis runs along the intersection of the floor and the right wall. The z-axis runs up from the floor toward the ceiling along the intersection of the two walls. You are situated in the first octant, and you can now imagine seven other rooms situated in the other seven octants (three on the same floor and four on the floor below), all connected by the common corner point O. Now if P is any point in space, let a be the (directed) distance from the yz-plane to P, let b be the distance from the xz-plane to P, and let c be the distance from the xy-plane to P. We represent the point P by the ordered triple 共a, b, c兲 of real numbers and we call a, b, and c the coordinates of P; a is the x-coordinate, b is the y-coordinate, and c is the z-coordinate. Thus, to locate the point 共a, b, c兲, we can start at the origin O and move a units along the x-axis, then b units parallel to the y-axis, and then c units parallel to the z-axis as in Figure 4.

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CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

The point P共a, b, c兲 determines a rectangular box as in Figure 5. If we drop a perpendicular from P to the xy-plane, we get a point Q with coordinates 共a, b, 0兲 called the projection of P on the xy-plane. Similarly, R共0, b, c兲 and S共a, 0, c兲 are the projections of P on the yz-plane and xz-plane, respectively. As numerical illustrations, the points 共⫺4, 3, ⫺5兲 and 共3, ⫺2, ⫺6兲 are plotted in Figure 6. z

z

z

3

(0, 0, c) R(0, b, c) S(a, 0, c)

0

_4 0

P(a, b, c)

_5

0

x

(_4, 3, _5)

(0, b, 0) x

y

y

x

(a, 0, 0)

3

_2

_6

y (3, _2, _6)

Q(a, b, 0)

FIGURE 5

FIGURE 6

ⱍ

The Cartesian product ⺢ ⫻ ⺢ ⫻ ⺢ 苷 兵共x, y, z兲 x, y, z 僆 ⺢其 is the set of all ordered triples of real numbers and is denoted by ⺢ 3. We have given a one-to-one correspondence between points P in space and ordered triples 共a, b, c兲 in ⺢ 3. It is called a threedimensional rectangular coordinate system. Notice that, in terms of coordinates, the first octant can be described as the set of points whose coordinates are all positive. In two-dimensional analytic geometry, the graph of an equation involving x and y is a curve in ⺢ 2. In three-dimensional analytic geometry, an equation in x, y, and z represents a surface in ⺢ 3. V EXAMPLE 1

(a) z 苷 3

What surfaces in ⺢ 3 are represented by the following equations? (b) y 苷 5

SOLUTION

ⱍ

(a) The equation z 苷 3 represents the set 兵共x, y, z兲 z 苷 3其, which is the set of all points in ⺢ 3 whose z-coordinate is 3. This is the horizontal plane that is parallel to the xy-plane and three units above it as in Figure 7(a). z

z

y 5

3 0 x

FIGURE 7

0 y

(a) z=3, a plane in R#

x

5

(b) y=5, a plane in R#

0

x

y

(c) y=5, a line in R@

(b) The equation y 苷 5 represents the set of all points in ⺢ 3 whose y-coordinate is 5. This is the vertical plane that is parallel to the xz-plane and five units to the right of it as M in Figure 7(b).

SECTION 12.1 THREE-DIMENSIONAL COORDINATE SYSTEMS

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767

NOTE When an equation is given, we must understand from the context whether it represents a curve in ⺢ 2 or a surface in ⺢ 3. In Example 1, y 苷 5 represents a plane in ⺢ 3, but of course y 苷 5 can also represent a line in ⺢ 2 if we are dealing with two-dimensional analytic geometry. See Figure 7(b) and (c). In general, if k is a constant, then x 苷 k represents a plane parallel to the yz-plane, y 苷 k is a plane parallel to the xz-plane, and z 苷 k is a plane parallel to the xy-plane. In Figure 5, the faces of the rectangular box are formed by the three coordinate planes x 苷 0 (the yz-plane), y 苷 0 (the xz-plane), and z 苷 0 (the xy-plane), and the planes x 苷 a, y 苷 b, and z 苷 c.

z

y 0 V EXAMPLE 2

Describe and sketch the surface in ⺢ 3 represented by the equation y 苷 x.

SOLUTION The equation represents the set of all points in ⺢ 3 whose x- and y-coordinates

ⱍ

x

are equal, that is, 兵共x, x, z兲 x 僆 ⺢, z 僆 ⺢其. This is a vertical plane that intersects the xy-plane in the line y 苷 x, z 苷 0. The portion of this plane that lies in the first octant is sketched in Figure 8. M

FIGURE 8

The plane y=x

The familiar formula for the distance between two points in a plane is easily extended to the following three-dimensional formula.

ⱍ

ⱍ

DISTANCE FORMULA IN THREE DIMENSIONS The distance P1 P2 between the

points P1共x 1, y1, z1 兲 and P2共x 2 , y2 , z2 兲 is

ⱍ P P ⱍ 苷 s共x 1

P™(¤, fi, z™)

ⱍP Aⱍ 苷 ⱍx 1

0 x

2

⫺ x1

ⱍ

ⱍ AB ⱍ 苷 ⱍ y

⫺ y1

2

ⱍ

ⱍ BP ⱍ 苷 ⱍ z 2

2

⫺ z1

ⱍ

Because triangles P1 BP2 and P1 AB are both right-angled, two applications of the Pythagorean Theorem give

B(¤, fi, z¡) A(¤, ›, z¡)

ⱍP P ⱍ

2

苷 P1 B

ⱍ

ⱍ

2

⫹ BP2

ⱍP Bⱍ

2

苷 P1 A

ⱍ

ⱍ

2

⫹ AB

1

y

FIGURE 9

⫺ x 1 兲2 ⫹ 共 y2 ⫺ y1 兲2 ⫹ 共z2 ⫺ z1 兲2

2

To see why this formula is true, we construct a rectangular box as in Figure 9, where P1 and P2 are opposite vertices and the faces of the box are parallel to the coordinate planes. If A共x 2 , y1, z1兲 and B共x 2 , y2 , z1兲 are the vertices of the box indicated in the figure, then

z P¡(⁄, ›, z¡)

2

and

2

1

ⱍ

ⱍ

2

ⱍ ⱍ

2

Combining these equations, we get

ⱍP P ⱍ 1

2

2

ⱍ ⱍ ⫹ ⱍ AB ⱍ ⫹ ⱍ BP ⱍ 苷 ⱍx ⫺ x ⱍ ⫹ ⱍy ⫺ y ⱍ ⫹ ⱍz 苷 P1 A 2

2

2

1

2

2

2

1

2

2

2

⫺ z1

ⱍ

2

苷 共x 2 ⫺ x 1 兲2 ⫹ 共 y2 ⫺ y1 兲2 ⫹ 共z2 ⫺ z1 兲2 Therefore

ⱍ P P ⱍ 苷 s共x 1

2

2

⫺ x 1 兲2 ⫹ 共 y2 ⫺ y1 兲2 ⫹ 共z2 ⫺ z1 兲2

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CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

EXAMPLE 3 The distance from the point P共2, ⫺1, 7兲 to the point Q共1, ⫺3, 5兲 is

ⱍ PQ ⱍ 苷 s共1 ⫺ 2兲

2

V EXAMPLE 4

z

⫹ 共⫺3 ⫹ 1兲2 ⫹ 共5 ⫺ 7兲2 苷 s1 ⫹ 4 ⫹ 4 苷 3

M

Find an equation of a sphere with radius r and center C共h, k, l兲.

SOLUTION By definition, a sphere is the set of all points P共x, y, z兲 whose distance from

P(x, y, z)

ⱍ ⱍ

C is r. (See Figure 10.) Thus P is on the sphere if and only if PC 苷 r. Squaring both sides, we have PC 2 苷 r 2 or

ⱍ ⱍ

r

共x ⫺ h兲2 ⫹ 共 y ⫺ k兲2 ⫹ 共z ⫺ l 兲2 苷 r 2

C(h, k, l)

M

The result of Example 4 is worth remembering. 0 x y

EQUATION OF A SPHERE An equation of a sphere with center C共h, k, l 兲 and

radius r is

FIGURE 10

共x ⫺ h兲2 ⫹ 共 y ⫺ k兲2 ⫹ 共z ⫺ l 兲2 苷 r 2 In particular, if the center is the origin O, then an equation of the sphere is x 2 ⫹ y 2 ⫹ z2 苷 r 2

EXAMPLE 5 Show that x 2 ⫹ y 2 ⫹ z 2 ⫹ 4x ⫺ 6y ⫹ 2z ⫹ 6 苷 0 is the equation of a

sphere, and find its center and radius. SOLUTION We can rewrite the given equation in the form of an equation of a sphere if we complete squares:

共x 2 ⫹ 4x ⫹ 4兲 ⫹ 共y 2 ⫺ 6y ⫹ 9兲 ⫹ 共z 2 ⫹ 2z ⫹ 1兲 苷 ⫺6 ⫹ 4 ⫹ 9 ⫹ 1 共x ⫹ 2兲2 ⫹ 共 y ⫺ 3兲2 ⫹ 共z ⫹ 1兲2 苷 8 Comparing this equation with the standard form, we see that it is the equation of a sphere with center 共⫺2, 3, ⫺1兲 and radius s8 苷 2s2 .

M

EXAMPLE 6 What region in ⺢ 3 is represented by the following inequalities?

1 艋 x 2 ⫹ y 2 ⫹ z2 艋 4

z艋0

SOLUTION The inequalities

z

1 艋 x 2 ⫹ y 2 ⫹ z2 艋 4 can be rewritten as 1 艋 sx 2 ⫹ y 2 ⫹ z 2 艋 2

0 1 2 x

FIGURE 11

y

so they represent the points 共x, y, z兲 whose distance from the origin is at least 1 and at most 2. But we are also given that z 艋 0, so the points lie on or below the xy-plane. Thus the given inequalities represent the region that lies between (or on) the spheres x 2 ⫹ y 2 ⫹ z 2 苷 1 and x 2 ⫹ y 2 ⫹ z 2 苷 4 and beneath (or on) the xy-plane. It is sketched in Figure 11. M

SECTION 12.1 THREE-DIMENSIONAL COORDINATE SYSTEMS

12.1

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EXERCISES

1. Suppose you start at the origin, move along the x-axis a

distance of 4 units in the positive direction, and then move downward a distance of 3 units. What are the coordinates of your position? 2. Sketch the points 共0, 5, 2兲, 共4, 0, ⫺1兲, 共2, 4, 6兲, and 共1, ⫺1, 2兲

on a single set of coordinate axes. 3. Which of the points P共6, 2, 3兲, Q共⫺5, ⫺1, 4兲, and R共0, 3, 8兲 is

closest to the xz-plane? Which point lies in the yz-plane? 4. What are the projections of the point (2, 3, 5) on the xy-, yz-,

and xz-planes? Draw a rectangular box with the origin and 共2, 3, 5兲 as opposite vertices and with its faces parallel to the coordinate planes. Label all vertices of the box. Find the length of the diagonal of the box.

15–18 Show that the equation represents a sphere, and find its center and radius. 15. x 2 ⫹ y 2 ⫹ z 2 ⫺ 6x ⫹ 4y ⫺ 2z 苷 11 16. x 2 ⫹ y 2 ⫹ z 2 ⫹ 8x ⫺ 6y ⫹ 2z ⫹ 17 苷 0 17. 2x 2 ⫹ 2y 2 ⫹ 2z 2 苷 8x ⫺ 24 z ⫹ 1 18. 4x 2 ⫹ 4y 2 ⫹ 4z 2 ⫺ 8x ⫹ 16y 苷 1

19. (a) Prove that the midpoint of the line segment from

P1共x 1, y1, z1 兲 to P2共x 2 , y2 , z2 兲 is

冉

5. Describe and sketch the surface in ⺢3 represented by the equa-

tion x ⫹ y 苷 2.

6. (a) What does the equation x 苷 4 represent in ⺢2 ? What does

it represent in ⺢3 ? Illustrate with sketches. (b) What does the equation y 苷 3 represent in ⺢3 ? What does z 苷 5 represent? What does the pair of equations y 苷 3, z 苷 5 represent? In other words, describe the set of points 共x, y, z兲 such that y 苷 3 and z 苷 5. Illustrate with a sketch.

7– 8 Find the lengths of the sides of the triangle PQR. Is it a right

triangle? Is it an isosceles triangle? 7. P共3, ⫺2, ⫺3兲, 8. P共2, ⫺1, 0兲,

Q共7, 0, 1兲, Q共4, 1, 1兲,

R共1, 2, 1兲 R共4, ⫺5, 4兲

9. Determine whether the points lie on straight line.

(a) A共2, 4, 2兲, B共3, 7, ⫺2兲, C共1, 3, 3兲 (b) D共0, ⫺5, 5兲, E共1, ⫺2, 4兲, F共3, 4, 2兲 10. Find the distance from 共3, 7, ⫺5兲 to each of the following.

(a) The xy-plane (c) The xz-plane (e) The y-axis

(b) The yz-plane (d) The x-axis (f) The z-axis

11. Find an equation of the sphere with center 共1, ⫺4, 3兲 and

radius 5. What is the intersection of this sphere with the xz-plane? 12. Find an equation of the sphere with center 共2, ⫺6, 4兲 and

radius 5. Describe its intersection with each of the coordinate planes. 13. Find an equation of the sphere that passes through the point

共4, 3, ⫺1兲 and has center 共3, 8, 1兲. 14. Find an equation of the sphere that passes through the origin

and whose center is 共1, 2, 3兲.

x 1 ⫹ x 2 y1 ⫹ y2 z1 ⫹ z2 , , 2 2 2

冊

(b) Find the lengths of the medians of the triangle with vertices A共1, 2, 3兲, B共⫺2, 0, 5兲, and C共4, 1, 5兲. 20. Find an equation of a sphere if one of its diameters has end-

points 共2, 1, 4兲 and 共4, 3, 10兲. 21. Find equations of the spheres with center 共2, ⫺3, 6兲 that touch

(a) the xy-plane, (b) the yz-plane, (c) the xz-plane. 22. Find an equation of the largest sphere with center (5, 4, 9) that

is contained in the first octant. 23–32 Describe in words the region of ⺢ 3 represented by the equa-

tion or inequality. 23. y 苷 ⫺4

24. x 苷 10

25. x ⬎ 3

26. y 艌 0

27. 0 艋 z 艋 6

28. z 2 苷 1

29. x 2 ⫹ y 2 ⫹ z 2 艋 3

30. x 苷 z

31. x ⫹ z 艋 9

32. x 2 ⫹ y 2 ⫹ z 2 ⬎ 2z

2

2

33–36 Write inequalities to describe the region. 33. The region between the yz-plane and the vertical plane x 苷 5 34. The solid cylinder that lies on or below the plane z 苷 8 and on

or above the disk in the xy-plane with center the origin and radius 2 35. The region consisting of all points between (but not on)

the spheres of radius r and R centered at the origin, where r ⬍ R 36. The solid upper hemisphere of the sphere of radius 2 centered

at the origin

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CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

37. The figure shows a line L 1 in space and a second line L 2 ,

which is the projection of L 1 on the xy-plane. (In other z

L¡

38. Consider the points P such that the distance from P to

P

A共⫺1, 5, 3兲 is twice the distance from P to B共6, 2, ⫺2兲. Show that the set of all such points is a sphere, and find its center and radius.

1 0 1

words, the points on L 2 are directly beneath, or above, the points on L 1.) (a) Find the coordinates of the point P on the line L 1. (b) Locate on the diagram the points A, B, and C, where the line L 1 intersects the xy-plane, the yz-plane, and the xz-plane, respectively.

39. Find an equation of the set of all points equidistant from the

L™

1

points A共⫺1, 5, 3兲 and B共6, 2, ⫺2兲. Describe the set. y

x

40. Find the volume of the solid that lies inside both of the spheres

x 2 ⫹ y 2 ⫹ z 2 ⫹ 4x ⫺ 2y ⫹ 4z ⫹ 5 苷 0 and

x 2 ⫹ y 2 ⫹ z2 苷 4

12.2 VECTORS

D B

u

v C A

FIGURE 1

Equivalent vectors

The term vector is used by scientists to indicate a quantity (such as displacement or velocity or force) that has both magnitude and direction. A vector is often represented by an arrow or a directed line segment. The length of the arrow represents the magnitude of the vector and the arrow points in the direction of the vector. We denote a vector by printing a letter in boldface 共v兲 or by putting an arrow above the letter 共 vl兲. For instance, suppose a particle moves along a line segment from point A to point B. The corresponding displacement vector v, shown in Figure 1, has initial point A (the tail) l and terminal point B (the tip) and we indicate this by writing v 苷 AB. Notice that the vecl tor u 苷 CD has the same length and the same direction as v even though it is in a different position. We say that u and v are equivalent (or equal) and we write u 苷 v. The zero vector, denoted by 0, has length 0. It is the only vector with no specific direction. COMBINING VECTORS

C B

A FIGURE 2

l Suppose a particle moves from A to B, so its displacement vector is AB. Then the particle l changes direction and moves from B to C, with displacement vector BC as in Figure 2. The combined effect of these displacements is that the particle has moved from A to C. The l l l resulting displacement vector AC is called the sum of AB and BC and we write l l l AC 苷 AB ⫹ BC In general, if we start with vectors u and v, we first move v so that its tail coincides with the tip of u and define the sum of u and v as follows. DEFINITION OF VECTOR ADDITION If u and v are vectors positioned so the initial point of v is at the terminal point of u, then the sum u ⫹ v is the vector from the initial point of u to the terminal point of v.

SECTION 12.2 VECTORS

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771

The definition of vector addition is illustrated in Figure 3. You can see why this definition is sometimes called the Triangle Law. u u+v

v v

u v+ v u+

v

u

u

FIGURE 4 The Parallelogram Law

FIGURE 3 The Triangle Law

In Figure 4 we start with the same vectors u and v as in Figure 3 and draw another copy of v with the same initial point as u. Completing the parallelogram, we see that u ⫹ v 苷 v ⫹ u. This also gives another way to construct the sum: If we place u and v so they start at the same point, then u ⫹ v lies along the diagonal of the parallelogram with u and v as sides. (This is called the Parallelogram Law.) V EXAMPLE 1

a

b

Draw the sum of the vectors a and b shown in Figure 5.

SOLUTION First we translate b and place its tail at the tip of a, being careful to draw a

copy of b that has the same length and direction. Then we draw the vector a ⫹ b [see Figure 6(a)] starting at the initial point of a and ending at the terminal point of the copy of b. Alternatively, we could place b so it starts where a starts and construct a ⫹ b by the Parallelogram Law as in Figure 6(b).

FIGURE 5

a

TEC Visual 12.2 shows how the Triangle and Parallelogram Laws work for various vectors a and b.

FIGURE 6

a

b a+b

a+b b

(a)

(b)

M

It is possible to multiply a vector by a real number c. (In this context we call the real number c a scalar to distinguish it from a vector.) For instance, we want 2v to be the same vector as v ⫹ v, which has the same direction as v but is twice as long. In general, we multiply a vector by a scalar as follows. DEFINITION OF SCALAR MULTIPLICATION If c is a scalar and v is a vector, then the

ⱍ ⱍ

2v

v

_v

_1.5v

FIGURE 7

Scalar multiples of v

1 2v

scalar multiple cv is the vector whose length is c times the length of v and whose direction is the same as v if c ⬎ 0 and is opposite to v if c ⬍ 0. If c 苷 0 or v 苷 0, then cv 苷 0. This definition is illustrated in Figure 7. We see that real numbers work like scaling factors here; that’s why we call them scalars. Notice that two nonzero vectors are parallel if they are scalar multiples of one another. In particular, the vector ⫺v 苷 共⫺1兲v has the same length as v but points in the opposite direction. We call it the negative of v. By the difference u ⫺ v of two vectors we mean u ⫺ v 苷 u ⫹ 共⫺v兲

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CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

So we can construct u ⫺ v by first drawing the negative of v, ⫺v, and then adding it to u by the Parallelogram Law as in Figure 8(a). Alternatively, since v ⫹ 共u ⫺ v兲 苷 u, the vector u ⫺ v, when added to v, gives u. So we could construct u ⫺ v as in Figure 8(b) by means of the Triangle Law.

v

u u-v

u-v

_v

v u

FIGURE 8

Drawing u-v

(a)

(b)

EXAMPLE 2 If a and b are the vectors shown in Figure 9, draw a ⫺ 2b.

SOLUTION We first draw the vector ⫺2b pointing in the direction opposite to b and twice as long. We place it with its tail at the tip of a and then use the Triangle Law to draw a ⫹ 共⫺2b兲 as in Figure 10. a

_2b a b

a-2b

FIGURE 9

FIGURE 10

M

COMPONENTS y

For some purposes it’s best to introduce a coordinate system and treat vectors algebraically. If we place the initial point of a vector a at the origin of a rectangular coordinate system, then the terminal point of a has coordinates of the form 共a1, a2 兲 or 共a1, a2, a3兲, depending on whether our coordinate system is two- or three-dimensional (see Figure 11). These coordinates are called the components of a and we write

(a¡, a™)

a O

x

a 苷 具 a 1, a 2 典

a=ka¡, a™l

or

a 苷 具a 1, a 2 , a 3 典

z (a¡, a™, a£)

a O y

x

We use the notation 具 a1, a2 典 for the ordered pair that refers to a vector so as not to confuse it with the ordered pair 共a1, a2 兲 that refers to a point in the plane. For instance, the vectors shown in Figure 12 are all equivalent to the vector l OP 苷 具3, 2典 whose terminal point is P共3, 2兲. What they have in common is that the terminal point is reached from the initial point by a displacement of three units to the right and two upward. We can think of all these geometric vectors as representations of the

a=ka¡, a™, a£l y

FIGURE 11

(4, 5) (1, 3)

0

FIGURE 12

Representations of the vector a=k3, 2l

P(3, 2)

x

SECTION 12.2 VECTORS

z

position vector of P P(a¡, a™, a£) O y A(x, y, z)

x

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773

l algebraic vector a 苷 具3, 2典 . The particular representation OP from the origin to the point P共3, 2兲 is called the position vector of the point P. l In three dimensions, the vector a 苷 OP 苷 具 a1, a2, a3 典 is the position vector of the l point P共a1, a2, a3兲. (See Figure 13.) Let’s consider any other representation AB of a, where the initial point is A共x 1, y1, z1 兲 and the terminal point is B共x 2 , y2 , z2 兲. Then we must have x 1 ⫹ a 1 苷 x 2, y1 ⫹ a 2 苷 y2, and z1 ⫹ a 3 苷 z2 and so a 1 苷 x 2 ⫺ x 1, a 2 苷 y2 ⫺ y1, and a 3 苷 z2 ⫺ z1. Thus we have the following result.

B(x+a¡, y+a™, z+a£)

Given the points A共x 1, y1, z1 兲 and B共x 2 , y2 , z2 兲, the vector a with represenl tation AB is a 苷 具 x 2 ⫺ x 1, y2 ⫺ y1, z2 ⫺ z1 典 1

FIGURE 13 Representations of a=ka¡, a™, a£l

V EXAMPLE 3 Find the vector represented by the directed line segment with initial point A共2, ⫺3, 4) and terminal point B共⫺2, 1, 1兲. l SOLUTION By (1), the vector corresponding to AB is

a 苷 具 ⫺2 ⫺ 2, 1 ⫺ 共⫺3兲, 1 ⫺ 4典 苷 具⫺4, 4, ⫺3典

M

The magnitude or length of the vector v is the length of any of its representations and is denoted by the symbol v or 储 v 储. By using the distance formula to compute the length of a segment OP, we obtain the following formulas.

ⱍ ⱍ

The length of the two-dimensional vector a 苷 具 a 1, a 2 典 is

ⱍ a ⱍ 苷 sa

2 1

⫹ a 22

The length of the three-dimensional vector a 苷 具 a 1, a 2 , a 3 典 is y

(a¡+b¡, a™+b™)

a+b

b™

b b¡ a a™ 0

a¡

a™ x

b¡

ⱍ a ⱍ 苷 sa

2 1

⫹ a 22 ⫹ a 32

How do we add vectors algebraically? Figure 14 shows that if a 苷 具a 1, a 2 典 and b 苷 具b 1, b 2 典 , then the sum is a ⫹ b 苷 具 a1 ⫹ b1, a2 ⫹ b2 典 , at least for the case where the components are positive. In other words, to add algebraic vectors we add their components. Similarly, to subtract vectors we subtract components. From the similar triangles in Figure 15 we see that the components of ca are ca1 and ca2. So to multiply a vector by a scalar we multiply each component by that scalar. If a 苷 具 a 1, a 2 典 and b 苷 具 b1, b2 典 , then

FIGURE 14

a ⫹ b 苷 具a 1 ⫹ b1, a 2 ⫹ b2 典

a ⫺ b 苷 具a 1 ⫺ b1, a 2 ⫺ b2 典 ca 苷 具 ca1, ca2 典

Similarly, for three-dimensional vectors, ca a

ca™

a™

具a 1, a 2 , a 3 典 ⫹ 具b1, b2 , b3 典 苷 具a 1 ⫹ b1, a 2 ⫹ b2 , a 3 ⫹ b3 典 具a 1, a 2 , a 3 典 ⫺ 具b1, b2 , b3 典 苷 具a 1 ⫺ b1, a 2 ⫺ b2 , a 3 ⫺ b3 典

a¡ FIGURE 15

ca¡

c 具a 1, a 2 , a 3 典 苷 具ca1, ca2 , ca3 典

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CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

ⱍ ⱍ

V EXAMPLE 4 If a 苷 具4, 0, 3典 and b 苷 具 2, 1, 5典 , find a and the vectors a  b, a  b, 3b, and 2a  5b.

ⱍ a ⱍ 苷 s4

SOLUTION

2

 0 2  32 苷 s25 苷 5

a  b 苷 具 4, 0, 3典  具 2, 1, 5典 苷 具4  共2兲, 0  1, 3  5典 苷 具2, 1, 8典 a  b 苷 具4, 0, 3典  具2, 1, 5典 苷 具 4  共2兲, 0  1, 3  5典 苷 具6, 1, 2 典 3b 苷 3具2, 1, 5典 苷 具 3共2兲, 3共1兲, 3共5兲典 苷 具6, 3, 15典 2a  5b 苷 2具4, 0, 3典  5 具2, 1, 5典 苷 具 8, 0, 6典  具10, 5, 25典 苷 具2, 5, 31典

M

We denote by V2 the set of all two-dimensional vectors and by V3 the set of all threedimensional vectors. More generally, we will later need to consider the set Vn of all n-dimensional vectors. An n-dimensional vector is an ordered n-tuple: Vectors in n dimensions are used to list various quantities in an organized way. For instance, the components of a six-dimensional vector

a 苷 具 a1, a 2, . . . , a n 典

N

p 苷 具 p1 , p2 , p3 , p4 , p5 , p6 典 might represent the prices of six different ingredients required to make a particular product. Four-dimensional vectors 具 x, y, z, t 典 are used in relativity theory, where the first three components specify a position in space and the fourth represents time.

where a1, a 2, . . . , a n are real numbers that are called the components of a. Addition and scalar multiplication are defined in terms of components just as for the cases n 苷 2 and n 苷 3. PROPERTIES OF VECTORS If a, b, and c are vectors in Vn and c and d are scalars,

then 1. a  b 苷 b  a

2. a  共b  c兲 苷 共a  b兲  c

3. a  0 苷 a

4. a  共a兲 苷 0

5. c共a  b兲 苷 ca  cb

6. 共c  d兲a 苷 ca  da

7. 共cd 兲a 苷 c共da兲

8. 1a 苷 a

These eight properties of vectors can be readily verified either geometrically or algebraically. For instance, Property 1 can be seen from Figure 4 (it’s equivalent to the Parallelogram Law) or as follows for the case n 苷 2: a  b 苷 具a 1, a 2 典  具b1, b2 典 苷 具a 1  b1, a 2  b2 典 苷 具b1  a 1, b2  a 2 典 苷 具b1, b2 典  具a 1, a 2 典 Q

苷ba

c

(a+b)+c =a+(b+c)

b

a+b b+c

P FIGURE 16

We can see why Property 2 (the associative law) is true by looking at Figure 16 and l applying the Triangle Law several times: The vector PQ is obtained either by first constructing a  b and then adding c or by adding a to the vector b  c. Three vectors in V3 play a special role. Let

a

i 苷 具1, 0, 0典

j 苷 具0, 1, 0 典

k 苷 具0, 0, 1典

SECTION 12.2 VECTORS

||||

775

These vectors i , j, and k are called the standard basis vectors. They have length 1 and point in the directions of the positive x-, y-, and z-axes. Similarly, in two dimensions we define i 苷 具1, 0典 and j 苷 具0, 1典 . (See Figure 17.) y

z

j

k

(0, 1)

0

x

i

j

i

(1, 0)

FIGURE 17

y

x

(a)

Standard basis vectors in V™ and V£

(b)

If a 苷 具a 1, a 2 , a 3 典 , then we can write a 苷 具a 1, a 2 , a 3 典 苷 具a 1, 0, 0典  具0, a 2 , 0典  具 0, 0, a 3 典 苷 a 1 具1, 0, 0典  a 2 具0, 1, 0典  a 3 具0, 0, 1典 y

2

(a¡, a™)

a

Thus any vector in V3 can be expressed in terms of i , j, and k. For instance,

a™ j

a¡i

0

具1, 2, 6典 苷 i  2j  6k

x

Similarly, in two dimensions, we can write

(a) a=a¡i+a™ j

3

z (a¡, a™, a£)

a 苷 具a1, a2 典 苷 a1 i  a2 j

See Figure 18 for the geometric interpretation of Equations 3 and 2 and compare with Figure 17.

a a£k

a¡i

y

x

a 苷 a1 i  a2 j  a3 k

a™ j (b) a=a¡i+a™j+a£k

EXAMPLE 5 If a 苷 i  2j  3k and b 苷 4i  7 k, express the vector 2a  3b in terms

of i , j, and k. SOLUTION Using Properties 1, 2, 5, 6, and 7 of vectors, we have

FIGURE 18

2a  3b 苷 2共i  2 j  3k兲  3共4i  7k兲 苷 2i  4 j  6k  12i  21k 苷 14i  4j  15k

M

A unit vector is a vector whose length is 1. For instance, i , j, and k are all unit vectors. In general, if a 苷 0, then the unit vector that has the same direction as a is 4

u苷

1 a a苷 a a

ⱍ ⱍ

ⱍ ⱍ

ⱍ ⱍ

In order to verify this, we let c 苷 1兾 a . Then u 苷 ca and c is a positive scalar, so u has the same direction as a. Also 1

ⱍ u ⱍ 苷 ⱍ ca ⱍ 苷 ⱍ c ⱍⱍ a ⱍ 苷 ⱍ a ⱍ ⱍ a ⱍ 苷 1

776

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CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

EXAMPLE 6 Find the unit vector in the direction of the vector 2i  j  2k.

SOLUTION The given vector has length

ⱍ 2i  j  2k ⱍ 苷 s2

2

 共1兲2  共2兲2 苷 s9 苷 3

so, by Equation 4, the unit vector with the same direction is 1 3

共2i  j  2k兲 苷 23 i  13 j  23 k

M

APPLICATIONS

Vectors are useful in many aspects of physics and engineering. In Chapter 13 we will see how they describe the velocity and acceleration of objects moving in space. Here we look at forces. A force is represented by a vector because it has both a magnitude (measured in pounds or newtons) and a direction. If several forces are acting on an object, the resultant force experienced by the object is the vector sum of these forces. 50°

32°

EXAMPLE 7 A 100-lb weight hangs from two wires as shown in Figure 19. Find the

tensions (forces) T1 and T2 in both wires and their magnitudes. T¡

T™

SOLUTION We first express T1 and T2 in terms of their horizontal and vertical components. From Figure 20 we see that

100 FIGURE 19

ⱍ ⱍ ⱍ ⱍ 苷 ⱍ T ⱍ cos 32 i  ⱍ T ⱍ sin 32 j

5

T1 苷  T1 cos 50 i  T1 sin 50 j

6

T2

2

2

.

The resultant T1  T2 of the tensions counterbalances the weight w and so we must have 50° T¡

32°

T1  T2 苷 w 苷 100 j

T™

Thus 50°

32° w

FIGURE 20

(ⱍ T1 ⱍ cos 50  ⱍ T2 ⱍ cos 32) i  (ⱍ T1 ⱍ sin 50  ⱍ T2 ⱍ sin 32) j 苷 100 j Equating components, we get

ⱍ ⱍ ⱍ ⱍ ⱍ T ⱍ sin 50  ⱍ T ⱍ sin 32 苷 100 Solving the first of these equations for ⱍ T ⱍ and substituting into the second, we get T cos 50 sin 32 苷 100 ⱍ T ⱍ sin 50  ⱍ ⱍ  T1 cos 50  T2 cos 32 苷 0 1

2

2

1

1

cos 32

So the magnitudes of the tensions are

ⱍT ⱍ 苷 1

and

100 ⬇ 85.64 lb sin 50  tan 32 cos 50

T cos 50 ⱍ T ⱍ 苷 ⱍ cosⱍ 32 ⬇ 64.91 lb 1

2

Substituting these values in (5) and (6), we obtain the tension vectors T1 ⬇ 55.05 i  65.60 j

T2 ⬇ 55.05 i  34.40 j

M

SECTION 12.2 VECTORS

12.2

777

EXERCISES

1. Are the following quantities vectors or scalars? Explain.

(a) (b) (c) (d)

||||

The cost of a theater ticket The current in a river The initial flight path from Houston to Dallas The population of the world

2. What is the relationship between the point (4, 7) and the

vector 具 4, 7 典 ? Illustrate with a sketch. 3. Name all the equal vectors in the parallelogram shown. A

B

9. A共1, 3兲,

B共2, 2兲

10. A共2, 1兲,

B共2, 3, 1兲

12. A共4, 0, 2兲,

11. A共0, 3, 1兲,

B共4, 2, 1兲

13–16 Find the sum of the given vectors and illustrate geometrically. 13. 具1, 4典 ,

具6, 2典

14. 具2, 1 典 ,

具5, 7典

15. 具0, 1, 2 典 ,

具 0, 0, 3 典

16. 具1, 0, 2典 ,

具 0, 4, 0典

ⱍ ⱍ

ⱍ

ⱍ

17–20 Find a  b, 2a  3b, a , and a  b . 17. a 苷 具5, 12典 ,

E

18. a 苷 4 i  j,

b 苷 具3, 6典

b 苷 i  2j

19. a 苷 i  2 j  3 k, D

B共0, 6兲

C

b 苷 2 i  j  5 k

20. a 苷 2 i  4 j  4 k,

b 苷 2j  k

4. Write each combination of vectors as a single vector.

l l (a) PQ  QR l l (c) QS  PS

l l (b) RP  PS l l l (d) RS  SP  PQ

21. 3 i  7 j

Q P

24. Find a vector that has the same direction as 具 2, 4, 2典 but has

length 6.

5. Copy the vectors in the figure and use them to draw the

(b) u  v (d) w  v  u u

v

25. If v lies in the first quadrant and makes an angle 兾3 with the

ⱍ ⱍ

positive x-axis and v 苷 4, find v in component form.

26. If a child pulls a sled through the snow on a level path with a

force of 50 N exerted at an angle of 38 above the horizontal, find the horizontal and vertical components of the force. w

27. A quarterback throws a football with angle of elevation 40 and

speed 60 ft兾s. Find the horizontal and vertical components of the velocity vector.

6. Copy the vectors in the figure and use them to draw the

following vectors. (a) a  b (c) 2a (e) 2a  b

a

22. 具4, 2, 4典

23. 8 i  j  4 k

S

R

following vectors. (a) u  v (c) v  w

21–23 Find a unit vector that has the same direction as the given

vector.

(b) a  b (d)  12 b (f) b  3a

28 –29 Find the magnitude of the resultant force and the angle it

makes with the positive x-axis. 28.

b

29.

y

0

y

20 lb

200 N 300 N

45° 30°

x

60° 0

x

16 lb

7–12 Find a vector a with representation given by the directed line l l segment AB. Draw AB and the equivalent representation starting at the origin. 7. A共2, 3兲,

B共2, 1兲

8. A共2, 2兲,

B共5, 3兲

30. The magnitude of a velocity vector is called speed. Suppose

that a wind is blowing from the direction N45 W at a speed of 50 km兾h. (This means that the direction from which the wind blows is 45 west of the northerly direction.) A pilot is steering

778

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CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

a plane in the direction N60 E at an airspeed (speed in still air) of 250 km兾h. The true course, or track, of the plane is the direction of the resultant of the velocity vectors of the plane and the wind. The ground speed of the plane is the magnitude of the resultant. Find the true course and the ground speed of the plane. 31. A woman walks due west on the deck of a ship at 3 mi兾h. The

ship is moving north at a speed of 22 mi兾h. Find the speed and direction of the woman relative to the surface of the water. 32. Ropes 3 m and 5 m in length are fastened to a holiday decora-

tion that is suspended over a town square. The decoration has a mass of 5 kg. The ropes, fastened at different heights, make angles of 52 and 40 with the horizontal. Find the tension in each wire and the magnitude of each tension. 52° 3 m

39. (a) Draw the vectors a 苷 具3, 2典 , b 苷 具2, 1典 , and

c 苷 具7, 1 典. (b) Show, by means of a sketch, that there are scalars s and t such that c 苷 sa  t b. (c) Use the sketch to estimate the values of s and t. (d) Find the exact values of s and t.

40. Suppose that a and b are nonzero vectors that are not parallel

and c is any vector in the plane determined by a and b. Give a geometric argument to show that c can be written as c 苷 sa  t b for suitable scalars s and t. Then give an argument using components. 41. If r 苷 具x, y, z 典 and r0 苷 具x 0 , y0 , z0 典 , describe the set of all

ⱍ

ⱍ

points 共x, y, z兲 such that r  r0 苷 1.

42. If r 苷 具x, y 典 , r1 苷 具 x 1, y1 典 , and r2 苷 具x 2 , y2 典 , describe the

40°

ⱍ

ⱍ ⱍ

ⱍ

set of all points 共x, y兲 such that r  r1  r  r2 苷 k, where k  r1  r2 .

ⱍ

5 m

ⱍ

43. Figure 16 gives a geometric demonstration of Property 2 of

vectors. Use components to give an algebraic proof of this fact for the case n 苷 2. 33. A clothesline is tied between two poles, 8 m apart. The line

is quite taut and has negligible sag. When a wet shirt with a mass of 0.8 kg is hung at the middle of the line, the midpoint is pulled down 8 cm. Find the tension in each half of the clothesline. 34. The tension T at each end of the chain has magnitude 25 N.

44. Prove Property 5 of vectors algebraically for the case n 苷 3.

Then use similar triangles to give a geometric proof. 45. Use vectors to prove that the line joining the midpoints of

two sides of a triangle is parallel to the third side and half its length.

What is the weight of the chain? 46. Suppose the three coordinate planes are all mirrored and a 37°

37°

35. Find the unit vectors that are parallel to the tangent line to the

parabola y 苷 x 2 at the point 共2, 4兲. 36. (a) Find the unit vectors that are parallel to the tangent line to

light ray given by the vector a 苷 具a 1, a 2 , a 3 典 first strikes the xz-plane, as shown in the figure. Use the fact that the angle of incidence equals the angle of reflection to show that the direction of the reflected ray is given by b 苷 具a 1, a 2 , a 3 典 . Deduce that, after being reflected by all three mutually perpendicular mirrors, the resulting ray is parallel to the initial ray. (American space scientists used this principle, together with laser beams and an array of corner mirrors on the moon, to calculate very precisely the distance from the earth to the moon.)

the curve y 苷 2 sin x at the point 共兾6, 1兲. (b) Find the unit vectors that are perpendicular to the tangent line. (c) Sketch the curve y 苷 2 sin x and the vectors in parts (a) and (b), all starting at 共兾6, 1兲.

z

37. If A, B, and C are the vertices of a triangle, find

l l l AB  BC  CA.

b

38. Let C be the point on the line segment AB that is twice

l l as far from B as it is from A. If a 苷 OA, b 苷 OB, and l 2 1 c 苷 OC, show that c 苷 3 a  3 b.

a x

y

SECTION 12.3 THE DOT PRODUCT

||||

779

12.3 THE DOT PRODUCT So far we have added two vectors and multiplied a vector by a scalar. The question arises: Is it possible to multiply two vectors so that their product is a useful quantity? One such product is the dot product, whose definition follows. Another is the cross product, which is discussed in the next section. 1 DEFINITION If a 苷 具 a 1, a 2 , a 3 典 and b 苷 具 b1, b2 , b3 典 , then the dot product of a and b is the number a ⴢ b given by

a ⴢ b 苷 a 1 b1  a 2 b2  a 3 b3 Thus, to find the dot product of a and b, we multiply corresponding components and add. The result is not a vector. It is a real number, that is, a scalar. For this reason, the dot product is sometimes called the scalar product (or inner product). Although Definition 1 is given for three-dimensional vectors, the dot product of two-dimensional vectors is defined in a similar fashion: 具a 1, a 2 典 ⴢ 具b1, b2 典 苷 a 1 b1  a 2 b2 V EXAMPLE 1

具 2, 4典 ⴢ 具 3, 1典 苷 2共3兲  4共1兲 苷 2 具1, 7, 4典 ⴢ 具6, 2,  12 典 苷 共1兲共6兲  7共2兲  4( 12 ) 苷 6 共i  2 j  3k兲 ⴢ 共2 j  k兲 苷 1共0兲  2共2兲  共3兲共1兲 苷 7

M

The dot product obeys many of the laws that hold for ordinary products of real numbers. These are stated in the following theorem. 2

PROPERTIES OF THE DOT PRODUCT If a, b, and c are vectors in V3 and c is a

scalar, then 1. a ⴢ a 苷 a 2 3. a ⴢ 共b  c兲 苷 a ⴢ b  a ⴢ c 5. 0 ⴢ a 苷 0

ⱍ ⱍ

2. a ⴢ b 苷 b ⴢ a 4. 共ca兲 ⴢ b 苷 c共a ⴢ b兲 苷 a ⴢ 共cb兲

These properties are easily proved using Definition 1. For instance, here are the proofs of Properties 1 and 3: 1. a ⴢ a 苷 a12  a 22  a 32 苷 a 2

ⱍ ⱍ

3. a ⴢ 共b  c兲 苷 具 a1, a2, a3 典 ⴢ 具b1  c1, b2  c2 , b3  c3 典

苷 a 1共b1  c1兲  a 2共b2  c2兲  a 3共b3  c3兲 苷 a 1 b1  a 1 c1  a 2 b2  a 2 c2  a 3 b3  a 3 c3 苷 共a 1 b1  a 2 b2  a 3 b3兲  共a 1 c1  a 2 c2  a 3 c3 兲 苷aⴢbaⴢc The proofs of the remaining properties are left as exercises.

M

The dot product a ⴢ b can be given a geometric interpretation in terms of the angle  between a and b, which is defined to be the angle between the representations of a and

780

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CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

z

B a-b

b 0 ¨ x

a

A

b that start at the origin, where 0    . In other words,  is the angle between the l l line segments OA and OB in Figure 1. Note that if a and b are parallel vectors, then  苷 0 or  苷 . The formula in the following theorem is used by physicists as the definition of the dot product. 3

THEOREM If  is the angle between the vectors a and b, then

y

ⱍ ⱍⱍ b ⱍ cos 

aⴢb苷 a

FIGURE 1

PROOF If we apply the Law of Cosines to triangle OAB in Figure 1, we get

ⱍ AB ⱍ

2

4

ⱍ

苷 OA

ⱍ

2

ⱍ

 OB

ⱍ

2

ⱍ

 2 OA

ⱍⱍ OB ⱍ cos 

(Observe that the Law of Cosines still applies in the limiting cases when  苷 0 or , or a 苷 0 or b 苷 0.) But OA 苷 a , OB 苷 b , and AB 苷 a  b , so Equation 4 becomes

ⱍ

ⱍ ⱍ ⱍⱍ

ⱍa  bⱍ

2

5

ⱍ ⱍ ⱍ

ⱍ ⱍ

苷 a

2

ⱍ ⱍ

 b

2

ⱍ ⱍ ⱍ

ⱍ

ⱍ ⱍⱍ b ⱍ cos 

2 a

Using Properties 1, 2, and 3 of the dot product, we can rewrite the left side of this equation as follows: a  b 2 苷 共a  b兲 ⴢ 共a  b兲

ⱍ

ⱍ

苷aⴢaaⴢbbⴢabⴢb

ⱍ ⱍ

苷 a

2

ⱍ ⱍ

 2a ⴢ b  b

2

Therefore Equation 5 gives

ⱍaⱍ

2

Thus or

ⱍ ⱍ

ⱍ ⱍ  2 ⱍ a ⱍⱍ b ⱍ cos  2a ⴢ b 苷 2 ⱍ a ⱍⱍ b ⱍ cos  a ⴢ b 苷 ⱍ a ⱍⱍ b ⱍ cos 

 2a ⴢ b  b

2

ⱍ ⱍ

苷 a

2

 b

2

M

EXAMPLE 2 If the vectors a and b have lengths 4 and 6, and the angle between them is

兾3, find a ⴢ b. SOLUTION Using Theorem 3, we have

ⱍ ⱍⱍ b ⱍ cos共兾3兲 苷 4 ⴢ 6 ⴢ

aⴢb苷 a

1 2

苷 12

M

The formula in Theorem 3 also enables us to find the angle between two vectors. 6

COROLLARY If  is the angle between the nonzero vectors a and b, then

cos  苷

V EXAMPLE 3

aⴢb a b

ⱍ ⱍⱍ ⱍ

Find the angle between the vectors a 苷 具2, 2, 1典 and b 苷 具 5, 3, 2典 .

SOLUTION Since

ⱍ a ⱍ 苷 s2

2

 2 2  共1兲2 苷 3

and

ⱍ b ⱍ 苷 s5

2

 共3兲2  2 2 苷 s38

SECTION 12.3 THE DOT PRODUCT

||||

781

and since a ⴢ b 苷 2共5兲  2共3兲  共1兲共2兲 苷 2 we have, from Corollary 6, cos  苷 So the angle between a and b is

aⴢb 2 苷 a b 3s38

ⱍ ⱍⱍ ⱍ

冉 冊

 苷 cos1

2 3s38

⬇ 1.46 共or 84兲

M

Two nonzero vectors a and b are called perpendicular or orthogonal if the angle between them is  苷 兾2. Then Theorem 3 gives

ⱍ ⱍⱍ b ⱍ cos共兾2兲 苷 0

aⴢb苷 a

and conversely if a ⴢ b 苷 0, then cos  苷 0, so  苷 兾2. The zero vector 0 is considered to be perpendicular to all vectors. Therefore we have the following method for determining whether two vectors are orthogonal. Two vectors a and b are orthogonal if and only if a ⴢ b 苷 0.

7

EXAMPLE 4 Show that 2i  2j  k is perpendicular to 5i  4 j  2 k.

SOLUTION Since

共2i  2j  k兲 ⴢ 共5i  4j  2k兲 苷 2共5兲  2共4兲  共1兲共2兲 苷 0 a

¨

a

a · b0

b

8

cos 苷

aⴢi a1 苷 a i a

ⱍ ⱍⱍ ⱍ

(This can also be seen directly from Figure 3.)

ⱍ ⱍ

782

||||

CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

Similarly, we also have cos  苷

9

a2 a

cos 苷

ⱍ ⱍ

a3 a

ⱍ ⱍ

By squaring the expressions in Equations 8 and 9 and adding, we see that cos 2  cos 2  cos 2 苷 1

10

We can also use Equations 8 and 9 to write

ⱍ ⱍ

ⱍ ⱍ

ⱍ ⱍ

a 苷 具 a 1, a 2 , a 3 典 苷 具 a cos , a cos , a cos 典

ⱍ ⱍ

苷 a 具cos , cos , cos 典 Therefore 1 a 苷 具 cos , cos , cos 典 a

ⱍ ⱍ

11

which says that the direction cosines of a are the components of the unit vector in the direction of a. EXAMPLE 5 Find the direction angles of the vector a 苷 具 1, 2, 3典 .

ⱍ ⱍ

SOLUTION Since a 苷 s1 2  2 2  3 2 苷 s14 , Equations 8 and 9 give

cos 苷 and so

冉 冊

苷 cos1

1 s14

1 s14

cos  苷

2 s14

冉 冊

 苷 cos1

⬇ 74

2 s14

cos 苷

3 s14

冉 冊

苷 cos1

⬇ 58

3 s14

⬇ 37 M

PROJECTIONS

l l Figure 4 shows representations PQ and PR of two vectors a and b with the same initial l point P. If S is the foot of the perpendicular from R to the line containing PQ, then the l vector with representation PS is called the vector projection of b onto a and is denoted by proja b. (You can think of it as a shadow of b.) R

TEC Visual 12.3B shows how Figure 4 changes when we vary a and b.

R b

b

a

a FIGURE 4

Vector projections

P

S proja b

Q S

P

Q

proja b

The scalar projection of b onto a (also called the component of b along a) is defined to be the signed magnitude of the vector projection, which is the number b cos , where

ⱍ ⱍ

SECTION 12.3 THE DOT PRODUCT

783

 is the angle between a and b. (See Figure 5.) This is denoted by compa b. Observe that it is negative if 兾2   . The equation

R

b

ⱍ ⱍⱍ b ⱍ cos  苷 ⱍ a ⱍ( ⱍ b ⱍ cos  )

aⴢb苷 a

a

¨

P

||||

Q

S 兩 b兩 cos ¨ = compa b

shows that the dot product of a and b can be interpreted as the length of a times the scalar projection of b onto a. Since

FIGURE 5

aⴢb

ⱍ b ⱍ cos  苷 ⱍ a ⱍ

Scalar projection

苷

a ⴢb a

ⱍ ⱍ

the component of b along a can be computed by taking the dot product of b with the unit vector in the direction of a. We summarize these ideas as follows.

Scalar projection of b onto a:

compa b 苷

Vector projection of b onto a:

proja b 苷

aⴢb a

ⱍ ⱍ

冉ⱍ ⱍ冊ⱍ ⱍ aⴢb a

a aⴢb 苷 a a a 2

ⱍ ⱍ

Notice that the vector projection is the scalar projection times the unit vector in the direction of a. V EXAMPLE 6 Find the scalar projection and vector projection of b 苷 具 1, 1, 2典 onto a 苷 具2, 3, 1典 .

ⱍ ⱍ

SOLUTION Since a 苷 s共2兲2  3 2  1 2 苷 s14 , the scalar projection of b onto a is

compa b 苷

aⴢb 共2兲共1兲  3共1兲  1共2兲 3 苷 苷 a s14 s14

ⱍ ⱍ

The vector projection is this scalar projection times the unit vector in the direction of a: proja b 苷

F

FIGURE 6

ⱍ ⱍ

Q D

ⱍ ⱍ

ⱍ ⱍ

W 苷 ( F cos ) D

S

P

冓

a 3 3 9 3 苷 a苷  , , a 14 7 14 14

冔

M

One use of projections occurs in physics in calculating work. In Section 6.4 we defined the work done by a constant force F in moving an object through a distance d as W 苷 Fd, but this applies only when the force is directed along the line of motion of the object. l Suppose, however, that the constant force is a vector F 苷 PR pointing in some other direction, as in Figure 6. If the force moves the object from P to Q, then the displacement l vector is D 苷 PQ. The work done by this force is defined to be the product of the component of the force along D and the distance moved:

R

¨

3 s14

But then, from Theorem 3, we have 12

ⱍ ⱍⱍ D ⱍ cos  苷 F ⴢ D

W苷 F

784

||||

CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

Thus the work done by a constant force F is the dot product F ⴢ D, where D is the displacement vector. EXAMPLE 7 A wagon is pulled a distance of 100 m along a horizontal path by a constant 35°

force of 70 N. The handle of the wagon is held at an angle of 35 above the horizontal. Find the work done by the force. SOLUTION If F and D are the force and displacement vectors, as pictured in Figure 7, then

the work done is

F 35°

ⱍ ⱍⱍ D ⱍ cos 35

WFⴢD F

D

 共70兲共100兲 cos 35 ⬇ 5734 Nm  5734 J

FIGURE 7

M

EXAMPLE 8 A force is given by a vector F  3i  4j  5k and moves a particle from

the point P共2, 1, 0兲 to the point Q共4, 6, 2兲. Find the work done. l SOLUTION The displacement vector is D  PQ  具2, 5, 2典 , so by Equation 12, the work done is W  F ⴢ D  具3, 4, 5典 ⴢ 具2, 5, 2典  6  20  10  36 If the unit of length is meters and the magnitude of the force is measured in newtons, then the work done is 36 joules.

12.3

M

EXERCISES

1. Which of the following expressions are meaningful? Which are

meaningless? Explain. (a) 共a ⴢ b兲 ⴢ c (c) a 共b ⴢ c兲 (e) a ⴢ b  c

ⱍ ⱍ

(b) 共a ⴢ b兲c (d) a ⴢ 共b  c兲 (f) a ⴢ 共b  c兲

ⱍ ⱍ

11–12 If u is a unit vector, find u ⴢ v and u ⴢ w. 11.

12. u

v

v w

2. Find the dot product of two vectors if their lengths are 6

and 13 and the angle between them is 兾4.

u

w

3–10 Find a ⴢ b. 3. a  具 2, 3 典 ,

b  具5, 12 典

4. a  具2, 3典 ,

b  具0.7, 1.2 典

5. a  具4, 1,

b  具 6, 3, 8 典

1

1 4

典,

6. a  具 s, 2s, 3s典 ,

b  具 t, t, 5t典

7. a  i  2 j  3 k , 8. a  4 j  3 k,

b  2i  4 j  6k

ⱍ ⱍ ⱍ b ⱍ  5 , the angle between a and b is 2兾3 ⱍ a ⱍ  3, ⱍ b ⱍ  s6 , the angle between a and b is 45

9. a  6, 10.

b  5i  9k

13. (a) Show that i ⴢ j  j ⴢ k  k ⴢ i  0.

(b) Show that i ⴢ i  j ⴢ j  k ⴢ k  1.

14. A street vendor sells a hamburgers, b hot dogs, and c soft

drinks on a given day. He charges $2 for a hamburger, $1.50 for a hot dog, and $1 for a soft drink. If A  具a, b, c 典 and P  具 2, 1.5, 1典 , what is the meaning of the dot product A ⴢ P ? 15–20 Find the angle between the vectors. (First find an exact

expression and then approximate to the nearest degree.) 15. a  具8, 6典 , 16. a  具 s3 , 1 典 ,

b  具s7 , 3 典 b  具0, 5 典

SECTION 12.3 THE DOT PRODUCT

17. a  具3, 1, 5典 , 18. a  具 4, 0, 2 典 , 19. a  j  k,

b  具2, 4, 3 典

39. a  2 i  j  4 k,

b  具 2, 1, 0典

40. a  i  j  k,

b  j  12 k

bijk

41. Show that the vector orth a b  b  proj a b is orthogonal to a.

b  4i  3k

(It is called an orthogonal projection of b.)

21–22 Find, correct to the nearest degree, the three angles of the

triangle with the given vertices. B共3, 6兲,

22. D共0, 1, 1兲,

E共2, 4, 3兲,

42. For the vectors in Exercise 36, find orth a b and illustrate by

drawing the vectors a, b, proj a b, and orth a b. 43. If a  具3, 0, 1 典 , find a vector b such that comp a b  2.

C共1, 4兲

44. Suppose that a and b are nonzero vectors.

F共1, 2, 1兲

(a) Under what circumstances is comp a b  comp b a? (b) Under what circumstances is proj a b  proj b a? 45. Find the work done by a force F  8 i  6 j  9 k that moves

23–24 Determine whether the given vectors are orthogonal,

an object from the point 共0, 10, 8兲 to the point 共6, 12, 20兲 along a straight line. The distance is measured in meters and the force in newtons.

parallel, or neither. 23. (a) a  具5, 3, 7 典 ,

b  具6, 8, 2 典 (b) a  具4, 6 典 , b  具3, 2 典 (c) a  i  2 j  5 k, b  3 i  4 j  k (d) a  2 i  6 j  4 k, b  3 i  9 j  6 k

46. A tow truck drags a stalled car along a road. The chain makes

an angle of 30 with the road and the tension in the chain is 1500 N. How much work is done by the truck in pulling the car 1 km?

24. (a) u  具3, 9, 6 典 ,

v  具4, 12, 8 典 (b) u  i  j  2 k, v  2 i  j  k (c) u  具a, b, c 典 , v  具b, a, 0 典

47. A sled is pulled along a level path through snow by a rope. A

30-lb force acting at an angle of 40 above the horizontal moves the sled 80 ft. Find the work done by the force.

25. Use vectors to decide whether the triangle with vertices

P共1, 3, 2兲, Q共2, 0, 4兲, and R共6, 2, 5兲 is right-angled. 26. For what values of b are the vectors 具6, b, 2 典 and 具b, b , b典 2

orthogonal? 27. Find a unit vector that is orthogonal to both i  j and i  k.

48. A boat sails south with the help of a wind blowing in the direc-

tion S36E with magnitude 400 lb. Find the work done by the wind as the boat moves 120 ft. 49. Use a scalar projection to show that the distance from a point

P1共x 1, y1兲 to the line ax  by  c  0 is

28. Find two unit vectors that make an angle of 60 with

ⱍ ax

 by1  c sa 2  b 2

v  具3, 4 典 .

1

29–33 Find the direction cosines and direction angles of the

vector. (Give the direction angles correct to the nearest degree.) 29. 具3, 4, 5 典

30. 具1, 2, 1典

31. 2 i  3 j  6 k

32. 2 i  j  2 k

33. 具c, c, c 典 ,

785

b  i  2 j  3k

20. a  i  2 j  2 k,

21. A共1, 0兲,

||||

where c 0

ⱍ

Use this formula to find the distance from the point 共2, 3兲 to the line 3x  4y  5  0. 50. If r  具x, y, z 典, a  具a 1, a 2 , a 3 典 , and b  具 b1, b2 , b3 典 , show

that the vector equation 共r  a兲 ⴢ 共r  b兲  0 represents a sphere, and find its center and radius.

51. Find the angle between a diagonal of a cube and one of its

edges. 34. If a vector has direction angles  兾4 and   兾3, find the

third direction angle .

35– 40 Find the scalar and vector projections of b onto a. 35. a  具3, 4典 , 36. a  具 1, 2典 ,

b  具 5, 0典 b  具 4, 1 典

37. a  具 3, 6, 2 典 , 38. a  具 2, 3, 6 典 ,

b  具1, 2, 3 典 b  具 5, 1, 4 典

52. Find the angle between a diagonal of a cube and a diagonal of

one of its faces. 53. A molecule of methane, CH 4 , is structured with the four hydro-

gen atoms at the vertices of a regular tetrahedron and the carbon atom at the centroid. The bond angle is the angle formed by the H— C—H combination; it is the angle between the lines that join the carbon atom to two of the hydrogen atoms. Show that the bond angle is about 109.5. [Hint: Take the vertices of the tetrahedron to be the points 共1, 0, 0兲, 共0, 1, 0兲 ,

786

||||

CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

共0, 0, 1兲, and 共1, 1, 1兲 as shown in the figure. Then the centroid is ( 12 , 12 , 12 ).]

57. Use Theorem 3 to prove the Cauchy-Schwarz Inequality:

ⱍa ⴢ bⱍ  ⱍaⱍⱍbⱍ

z

58. The Triangle Inequality for vectors is

H

ⱍa  bⱍ  ⱍaⱍ  ⱍbⱍ C

H

(a) Give a geometric interpretation of the Triangle Inequality. (b) Use the Cauchy-Schwarz Inequality from Exercise 57 to prove the Triangle Inequality. [Hint: Use the fact that a  b 2  共a  b兲  共a  b兲 and use Property 3 of the dot product.]

H y

x

ⱍ ⱍ

ⱍ

H

ⱍ

59. The Parallelogram Law states that

ⱍ ⱍ

54. If c  a b  b a, where a, b, and c are all nonzero

ⱍa  bⱍ

2

vectors, show that c bisects the angle between a and b. 55. Prove Properties 2, 4, and 5 of the dot product (Theorem 2).

ⱍ

 ab

ⱍ

2

ⱍ ⱍ

2 a

2

ⱍ ⱍ

2 b

2

(a) Give a geometric interpretation of the Parallelogram Law. (b) Prove the Parallelogram Law. (See the hint in Exercise 58.)

56. Suppose that all sides of a quadrilateral are equal in length and

60. Show that if u  v and u  v are orthogonal, then the vectors

opposite sides are parallel. Use vector methods to show that the diagonals are perpendicular.

u and v must have the same length.

12.4 THE CROSS PRODUCT The cross product a b of two vectors a and b, unlike the dot product, is a vector. For this reason it is also called the vector product. Note that a b is defined only when a and b are three-dimensional vectors. 1 DEFINITION If a  具 a 1, a 2 , a 3 典 and b  具b1, b2 , b3 典 , then the cross product of a and b is the vector

a b  具a 2 b3  a 3 b2 , a 3 b1  a 1 b3 , a 1 b2  a 2 b1 典 This may seem like a strange way of defining a product. The reason for the particular form of Definition 1 is that the cross product defined in this way has many useful properties, as we will soon see. In particular, we will show that the vector a b is perpendicular to both a and b. In order to make Definition 1 easier to remember, we use the notation of determinants. A determinant of order 2 is defined by

冟 冟 冟 冟 a c

2 6

For example,

b  ad  bc d

1  2共4兲  1共6兲  14 4

A determinant of order 3 can be defined in terms of second-order determinants as follows: 2

ⱍ ⱍ a1 b1 c1

a2 b2 c2

冟

a3 b2 b3  a1 c2 c3

冟 冟

b3 b1  a2 c3 c1

冟 冟

b3 b1  a3 c3 c1

b2 c2

冟

SECTION 12.4 THE CROSS PRODUCT

||||

787

Observe that each term on the right side of Equation 2 involves a number a i in the first row of the determinant, and a i is multiplied by the second-order determinant obtained from the left side by deleting the row and column in which a i appears. Notice also the minus sign in the second term. For example,

ⱍ ⱍ 1 3 5

冟 冟 冟

1 0 1 1 4 2

2 0 4

1 3 2 2 5

冟

冟

1 3 0  共1兲 2 5 4

冟

 1共0  4兲  2共6  5兲  共1兲共12  0兲  38

If we now rewrite Definition 1 using second-order determinants and the standard basis vectors i , j, and k, we see that the cross product of the vectors a  a 1 i  a 2 j  a 3 k and b  b 1 i  b 2 j  b 3 k is a b

3

冟

冟 冟

a2 b2

a3 a1 i b3 b1

冟 冟

a3 a1 j b3 b1

冟

a2 k b2

In view of the similarity between Equations 2 and 3, we often write

ⱍ ⱍ

i j a b  a1 a2 b1 b2

4

k a3 b3

Although the first row of the symbolic determinant in Equation 4 consists of vectors, if we expand it as if it were an ordinary determinant using the rule in Equation 2, we obtain Equation 3. The symbolic formula in Equation 4 is probably the easiest way of remembering and computing cross products. V EXAMPLE 1

If a  具1, 3, 4典 and b  具 2, 7, 5典 , then

ⱍ冟 冟 ⱍ 冟

i a b 1 2 

3 7

j 3 7

k 4 5

4 1 i 5 2

冟 冟 冟

4 1 j 5 2

3 k 7

 共15  28兲 i  共5  8兲 j  共7  6兲 k  43i  13j  k V EXAMPLE 2

M

Show that a a  0 for any vector a in V3.

SOLUTION If a  具 a 1, a 2 , a 3 典 , then

ⱍ ⱍ

i j a a  a1 a2 a1 a2

k a3 a3

 共a 2 a 3  a 3 a 2兲 i  共a 1 a 3  a 3 a 1兲 j  共a 1 a 2  a 2 a 1兲 k  0i  0j  0k  0

M

788

||||

CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

One of the most important properties of the cross product is given by the following theorem. 5

THEOREM The vector a b is orthogonal to both a and b.

PROOF In order to show that a b is orthogonal to a, we compute their dot product as follows:

共a b兲 ⴢ a 

冟

a2 b2

冟 冟

a3 a1 a1  b3 b1

冟 冟

a3 a1 a2  b3 b1

冟

a2 a3 b2

 a 1共a 2 b3  a 3 b2 兲  a 2共a 1 b3  a 3 b1 兲  a 3共a 1 b2  a 2 b1 兲  a 1 a 2 b3  a 1 b2 a 3  a 1 a 2 b3  b1 a 2 a 3  a 1 b2 a 3  b1 a 2 a 3 0 A similar computation shows that 共a b兲 ⴢ b  0. Therefore a b is orthogonal to both a and b.

axb

a

¨

b

FIGURE 1

TEC Visual 12.4 shows how a b changes as b changes.

M

If a and b are represented by directed line segments with the same initial point (as in Figure 1), then Theorem 5 says that the cross product a b points in a direction perpendicular to the plane through a and b. It turns out that the direction of a b is given by the right-hand rule: If the fingers of your right hand curl in the direction of a rotation (through an angle less than 180) from a to b, then your thumb points in the direction of a b. Now that we know the direction of the vector a b, the remaining thing we need to complete its geometric description is its length a b . This is given by the following theorem.

ⱍ

6

ⱍ

THEOREM If is the angle between a and b (so 0   ), then

ⱍ a b ⱍ  ⱍ a ⱍⱍ b ⱍ sin PROOF From the definitions of the cross product and length of a vector, we have

ⱍa bⱍ

2

 共a 2 b3  a 3 b2兲2  共a 3 b1  a 1 b3兲2  共a 1 b2  a 2 b1兲2  a 22b 32  2a 2 a 3 b2 b3  a 32 b 22  a 32b12  2a 1 a 3 b1 b3  a12 b 23  a12 b 22  2a 1 a 2 b1 b2  a 22b12  共a12  a 22  a 32 兲共b 12  b 22  b 32 兲  共a 1 b1  a 2 b2  a 3 b3 兲2

ⱍ ⱍ ⱍ b ⱍ  共a ⴢ b兲  ⱍ a ⱍ ⱍ b ⱍ  ⱍ a ⱍ ⱍ b ⱍ cos  ⱍ a ⱍ ⱍ b ⱍ 共1  cos 兲  ⱍ a ⱍ ⱍ b ⱍ sin  a

2

2

2

2

2

2

2

2

2

2

2

2

(by Theorem 12.3.3)

2

2

Taking square roots and observing that ssin 2  sin because sin  0 when 0   , we have a b  a b sin

ⱍ

Geometric characterization of a b

ⱍ ⱍ ⱍⱍ ⱍ

M

Since a vector is completely determined by its magnitude and direction, we can now say that a b is the vector that is perpendicular to both a and b, whose orientation is deter-

SECTION 12.4 THE CROSS PRODUCT

||||

789

ⱍ ⱍⱍ b ⱍ sin . In fact, that is exactly how

mined by the right-hand rule, and whose length is a physicists define a b. 7

COROLLARY Two nonzero vectors a and b are parallel if and only if

a b0 PROOF Two nonzero vectors a and b are parallel if and only if  0 or . In either case sin  0, so a b  0 and therefore a b  0. M

ⱍ

b

兩 b 兩 sin ¨

¨ FIGURE 2

ⱍ

The geometric interpretation of Theorem 6 can be seen by looking at Figure 2. If a and b are represented by directed line segments with the same initial point, then they determine a parallelogram with base a , altitude b sin , and area

ⱍ ⱍ

ⱍ ⱍ

ⱍ ⱍ (ⱍ b ⱍ sin )  ⱍ a b ⱍ

A a

a

Thus we have the following way of interpreting the magnitude of a cross product. The length of the cross product a b is equal to the area of the parallelogram determined by a and b. EXAMPLE 3 Find a vector perpendicular to the plane that passes through the points

P共1, 4, 6兲, Q共2, 5, 1兲, and R共1, 1, 1兲. l l l l SOLUTION The vector PQ PR is perpendicular to both PQ and PR and is therefore perpendicular to the plane through P, Q, and R. We know from (12.2.1) that l PQ  共2  1兲 i  共5  4兲 j  共1  6兲 k  3i  j  7k l PR  共1  1兲 i  共1  4兲 j  共1  6兲 k  5 j  5k We compute the cross product of these vectors:

ⱍ

i l l PQ PR  3 0

j 1 5

k 7 5

ⱍ

 共5  35兲 i  共15  0兲 j  共15  0兲 k  40 i  15 j  15k So the vector 具40, 15, 15典 is perpendicular to the given plane. Any nonzero scalar multiple of this vector, such as 具8, 3, 3典 , is also perpendicular to the plane.

M

EXAMPLE 4 Find the area of the triangle with vertices P共1, 4, 6兲, Q共2, 5, 1兲,

and R共1, 1, 1兲.

l

l

SOLUTION In Example 3 we computed that PQ PR  具40, 15, 15典 . The area of the

parallelogram with adjacent sides PQ and PR is the length of this cross product: l l PR ⱍ  s共40兲 ⱍ PQ

2

 共15兲2  15 2  5s82

5 The area A of the triangle PQR is half the area of this parallelogram, that is, 2 s82 .

M

790

||||

CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

If we apply Theorems 5 and 6 to the standard basis vectors i , j, and k using  兾2, we obtain i jk

j ki

k ij

j i  k

k j  i

i k  j

Observe that i jj i | Thus the cross product is not commutative. Also

i 共i j兲  i k  j whereas 共i i兲 j  0 j  0 | So the associative law for multiplication does not usually hold; that is, in general,

共a b兲 c  a 共b c兲 However, some of the usual laws of algebra do hold for cross products. The following theorem summarizes the properties of vector products. 8

THEOREM If a, b, and c are vectors and c is a scalar, then

1. a b  b a 2. (ca) b  c(a b)  a (cb) 3. a (b  c)  a b  a c 4. (a  b) c  a c  b c 5. a ⴢ 共b c兲  共a b兲 ⴢ c 6. a 共b c兲  共a ⴢ c兲b  共a ⴢ b兲c

These properties can be proved by writing the vectors in terms of their components and using the definition of a cross product. We give the proof of Property 5 and leave the remaining proofs as exercises. PROOF OF PROPERTY 5 If a  具 a 1, a 2 , a 3 典 , b  具b1, b2 , b3 典 , and c  具 c1, c2 , c3 典 , then 9

a ⴢ 共b c兲  a 1共b2 c3  b3 c2兲  a 2共b3 c1  b1 c3兲  a 3共b1 c2  b2 c1兲  a 1 b2 c3  a 1 b3 c2  a 2 b3 c1  a 2 b1 c3  a 3 b1 c2  a 3 b2 c1  共a 2 b3  a 3 b2 兲c1  共a 3 b1  a 1 b3 兲c2  共a 1 b2  a 2 b1 兲c3  共a b兲 ⴢ c

M

TRIPLE PRODUCTS

The product a ⴢ 共b c兲 that occurs in Property 5 is called the scalar triple product of the vectors a, b, and c. Notice from Equation 9 that we can write the scalar triple product as a determinant: 10

ⱍ ⱍ

a1 a ⴢ 共b c兲  b1 c1

a2 b2 c2

a3 b3 c3

SECTION 12.4 THE CROSS PRODUCT

||||

791

The geometric significance of the scalar triple product can be seen by considering the parallelepiped determined by the vectors a, b, and c. (See Figure 3.) The area of the base parallelogram is A  b c . If is the angle between a and b c, then the height h of the parallelepiped is h  a cos . (We must use cos instead of cos in case

兾2.) Therefore the volume of the parallelepiped is

bxc

ⱍ

h ¨ a c

ⱍ ⱍ ⱍⱍ

ⱍ

ⱍ

b

V  Ah  b c

FIGURE 3

ⱍ

ⱍ

ⱍⱍ a ⱍⱍ cos ⱍ  ⱍ a ⴢ 共b c兲 ⱍ

Thus we have proved the following formula. 11 The volume of the parallelepiped determined by the vectors a, b, and c is the magnitude of their scalar triple product:

ⱍ

V  a ⴢ 共b c兲

ⱍ

If we use the formula in (11) and discover that the volume of the parallelepiped determined by a, b, and c is 0, then the vectors must lie in the same plane; that is, they are coplanar. Use the scalar triple product to show that the vectors a  具1, 4, 7典 , b  具2, 1, 4典 , and c  具 0, 9, 18典 are coplanar. V EXAMPLE 5

SOLUTION We use Equation 10 to compute their scalar triple product:

ⱍ冟 冟 ⱍ 冟

1 a ⴢ 共b c兲  2 0 1

4 1 9

1 9

7 4 18

4 2 4 18 0

冟 冟

4 2 7 18 0

1 9

冟

 1共18兲  4共36兲  7共18兲  0 Therefore, by (11), the volume of the parallelepiped determined by a, b, and c is 0. This means that a, b, and c are coplanar. M The product a 共b c兲 that occurs in Property 6 is called the vector triple product of a, b, and c. Property 6 will be used to derive Kepler’s First Law of planetary motion in Chapter 13. Its proof is left as Exercise 46. TORQUE

The idea of a cross product occurs often in physics. In particular, we consider a force F acting on a rigid body at a point given by a position vector r. (For instance, if we tighten a bolt by applying a force to a wrench as in Figure 4, we produce a turning effect.) The torque ␶ (relative to the origin) is defined to be the cross product of the position and force vectors

␶

r ¨ F FIGURE 4

␶r F and measures the tendency of the body to rotate about the origin. The direction of the torque vector indicates the axis of rotation. According to Theorem 6, the magnitude of the

792

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CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

torque vector is

ⱍ ␶ ⱍ  ⱍ r F ⱍ  ⱍ r ⱍⱍ F ⱍ sin where is the angle between the position and force vectors. Observe that the only component of F that can cause a rotation is the one perpendicular to r, that is, F sin . The magnitude of the torque is equal to the area of the parallelogram determined by r and F.

ⱍ ⱍ

EXAMPLE 6 A bolt is tightened by applying a 40-N force to a 0.25-m wrench as shown

in Figure 5. Find the magnitude of the torque about the center of the bolt. SOLUTION The magnitude of the torque vector is 75° 0.25 m

ⱍ ␶ ⱍ  ⱍ r F ⱍ  ⱍ r ⱍⱍ F ⱍ sin 75  共0.25兲共40兲 sin 75

40 N

 10 sin 75 ⬇ 9.66 Nm If the bolt is right-threaded, then the torque vector itself is

␶  ⱍ ␶ ⱍ n ⬇ 9.66 n where n is a unit vector directed down into the page.

FIGURE 5

12.4

M

EXERCISES

1–7 Find the cross product a b and verify that it is orthogonal

to both a and b. 1. a  具6, 0, 2 典 ,

b  具0, 8, 0 典

2. a  具 1, 1, 1典 ,

b  具2, 4, 6 典

3. a  i  3 j  2 k, 4. a  j  7 k,

7. a  具t, t , t 典 ,

14.

15. |u|=6

b  i  5 k

b  2i  j  2k 1

6. a  i  e t j  et k, 3

ⱍ

|u|=5

|v|=8

60°

150°

|v|=10

b  2i  j  4k

5. a  i  j  k,

2

ⱍ

14 –15 Find u v and determine whether u v is directed into the page or out of the page.

1

b  2 i  e t j  et k

b  具1, 2t, 3t 典 2

16. The figure shows a vector a in the xy-plane and a vector b in

ⱍ ⱍ

ⱍ ⱍ

the direction of k. Their lengths are a  3 and b  2. (a) Find a b . (b) Use the right-hand rule to decide whether the components of a b are positive, negative, or 0.

ⱍ

ⱍ

8. If a  i  2 k and b  j  k, find a b. Sketch a, b, and

z

a b as vectors starting at the origin.

9–12 Find the vector, not with determinants, but by using proper-

b

ties of cross products. 9. 共i j兲 k 11. 共 j  k兲 共k  i兲

10. k 共i  2 j兲

a x

12. 共i  j兲 共i  j兲

13. State whether each expression is meaningful. If not, explain

why. If so, state whether it is a vector or a scalar. (a) a ⴢ 共b c兲 (b) a 共b ⴢ c兲 (c) a 共b c兲 (d) 共a ⴢ b兲 c (e) 共a ⴢ b兲 共c ⴢ d兲 (f) 共a b兲 ⴢ 共c d兲

y

17. If a  具1, 2, 1 典 and b  具0, 1, 3典 , find a b and b a. 18. If a  具3, 1, 2 典 , b  具1, 1, 0 典 , and c  具 0, 0, 4典 , show

that a 共b c兲  共a b兲 c.

19. Find two unit vectors orthogonal to both 具1, 1, 1典 and

具 0, 4, 4典 .

SECTION 12.4 THE CROSS PRODUCT

20. Find two unit vectors orthogonal to both i  j  k

and 2 i  k.

||||

793

40. Find the magnitude of the torque about P if a 36-lb force is

applied as shown.

21. Show that 0 a  0  a 0 for any vector a in V3 .

4 ft P

22. Show that 共a b兲 ⴢ b  0 for all vectors a and b in V3 . 23. Prove Property 1 of Theorem 8.

4 ft

24. Prove Property 2 of Theorem 8. 25. Prove Property 3 of Theorem 8. 30° 36 lb

26. Prove Property 4 of Theorem 8. 27. Find the area of the parallelogram with vertices A共2, 1兲,

B共0, 4兲, C共4, 2兲, and D共2, 1兲.

41. A wrench 30 cm long lies along the positive y-axis and grips a

28. Find the area of the parallelogram with vertices K共1, 2, 3兲,

L共1, 3, 6兲, M共3, 8, 6兲, and N共3, 7, 3兲. 29–32 (a) Find a nonzero vector orthogonal to the plane through

the points P, Q, and R, and (b) find the area of triangle PQR. 29. P共1, 0, 0兲,

Q共0, 2, 0兲,

30. P共2, 1, 5兲,

Q共1, 3, 4兲,

R共0, 0, 3兲

31. P共0, 2, 0兲,

Q共4, 1, 2兲,

32. P共1, 3, 1兲,

Q共0, 5, 2兲,

R共3, 0, 6兲

bolt at the origin. A force is applied in the direction 具 0, 3, 4典 at the end of the wrench. Find the magnitude of the force needed to supply 100 Nm of torque to the bolt. 42. Let v  5j and let u be a vector with length 3 that starts at

the origin and rotates in the xy -plane. Find the maximum and minimum values of the length of the vector u v. In what direction does u v point? 43. (a) Let P be a point not on the line L that passes through the

R共5, 3, 1兲

points Q and R. Show that the distance d from the point P to the line L is

R共4, 3, 1兲

d

33–34 Find the volume of the parallelepiped determined by the

vectors a, b, and c. 33. a  具 6, 3, 1典 ,

b  具0, 1, 2 典 ,

c  具4, 2, 5典

34. a  i  j  k,

b  i  j  k,

c  i  j  k

35–36 Find the volume of the parallelepiped with adjacent edges

PQ, PR, and PS. 35. P共2, 0, 1兲, 36. P共3, 0, 1兲,

Q共4, 1, 0兲,

R共3, 1, 1兲,

S共2, 2, 2兲

Q共1, 2, 5兲,

R共5, 1, 1兲,

S共0, 4, 2兲

37. Use the scalar triple product to verify that the vectors

u  i  5 j  2 k, v  3 i  j, and w  5 i  9 j  4 k are coplanar. 38. Use the scalar triple product to determine whether the points

A共1, 3, 2兲, B共3, 1, 6兲, C共5, 2, 0兲, and D共3, 6, 4兲 lie in the same plane.

l l where a  QR and b  QP. (b) Use the formula in part (a) to find the distance from the point P共1, 1, 1兲 to the line through Q共0, 6, 8兲 and R共1, 4, 7兲. 44. (a) Let P be a point not on the plane that passes through the

points Q, R, and S. Show that the distance d from P to the plane is 共 a b兲 ⴢ c d a b

ⱍ

ⱍ

70°

46. Prove Property 6 of Theorem 8, that is,

a 共b c兲  共a ⴢ c兲b  共a ⴢ b兲c 47. Use Exercise 46 to prove that

a 共b c兲  b 共c a兲  c 共a b兲  0

共a b兲 ⴢ 共c d兲  10°

P

ⱍ

45. Prove that 共a  b兲 共a  b兲  2共a b兲.

48. Prove that 60 N

ⱍ

l l l where a  QR, b  QS, and c  QP. (b) Use the formula in part (a) to find the distance from the point P共2, 1, 4兲 to the plane through the points Q共1, 0, 0兲, R共0, 2, 0兲, and S共0, 0, 3兲.

39. A bicycle pedal is pushed by a foot with a 60-N force as

shown. The shaft of the pedal is 18 cm long. Find the magnitude of the torque about P.

ⱍa bⱍ ⱍaⱍ

49. Suppose that a  0.

冟

aⴢc aⴢd

bⴢc bⴢd

(a) If a ⴢ b  a ⴢ c, does it follow that b  c ?

冟

794

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CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

(b) If a ⫻ b 苷 a ⫻ c, does it follow that b 苷 c ? (c) If a ⴢ b 苷 a ⴢ c and a ⫻ b 苷 a ⫻ c, does it follow that b 苷 c ? 50. If v1, v2, and v3 are noncoplanar vectors, let

k1 苷

v2 ⫻ v3 v1 ⴢ 共v2 ⫻ v3 兲 k3 苷

k2 苷

v3 ⫻ v1 v1 ⴢ 共v2 ⫻ v3 兲

v1 ⫻ v2 v1 ⴢ 共v2 ⫻ v3 兲

D I S COV E RY PROJECT

(These vectors occur in the study of crystallography. Vectors of the form n1 v1 ⫹ n 2 v2 ⫹ n3 v3 , where each n i is an integer, form a lattice for a crystal. Vectors written similarly in terms of k1, k 2 , and k 3 form the reciprocal lattice.) (a) Show that k i is perpendicular to vj if i 苷 j. (b) Show that k i ⴢ vi 苷 1 for i 苷 1, 2, 3. 1 (c) Show that k1 ⴢ 共k 2 ⫻ k 3 兲 苷 . v1 ⴢ 共v2 ⫻ v3 兲

THE GEOMETRY OF A TETRAHEDRON A tetrahedron is a solid with four vertices, P, Q, R, and S, and four triangular faces as shown in the figure.

P

1. Let v1 , v2 , v3 , and v4 be vectors with lengths equal to the areas of the faces opposite the vertices P, Q, R, and S, respectively, and directions perpendicular to the respective faces and pointing outward. Show that v1 ⫹ v2 ⫹ v3 ⫹ v4 苷 0 S R

Q

2. The volume V of a tetrahedron is one-third the distance from a vertex to the opposite face, times the area of that face. (a) Find a formula for the volume of a tetrahedron in terms of the coordinates of its vertices P, Q, R, and S. (b) Find the volume of the tetrahedron whose vertices are P共1, 1, 1兲, Q共1, 2, 3兲, R共1, 1, 2兲, and S共3, ⫺1, 2兲. 3. Suppose the tetrahedron in the figure has a trirectangular vertex S. (This means that the three angles at S are all right angles.) Let A, B, and C be the areas of the three faces that meet at S, and let D be the area of the opposite face PQR. Using the result of Problem 1, or otherwise, show that D 2 苷 A2 ⫹ B 2 ⫹ C 2 (This is a three-dimensional version of the Pythagorean Theorem.)

12.5 EQUATIONS OF LINES AND PLANES A line in the xy-plane is determined when a point on the line and the direction of the line (its slope or angle of inclination) are given. The equation of the line can then be written using the point-slope form. Likewise, a line L in three-dimensional space is determined when we know a point P0共x 0 , y0 , z0兲 on L and the direction of L. In three dimensions the direction of a line is conveniently described by a vector, so we let v be a vector parallel to L. Let P共x, y, z兲 be an arbitrary point on L and let r0 and r be the position vectors of P0 and P (that is, they have representations OP A0 and OP A). If a is the vector with representation P A, 0 P as in Figure 1, then the Triangle Law for vector addition gives r 苷 r0 ⫹ a. But, since a and v are parallel vectors, there is a scalar t such that a 苷 t v. Thus

z

P¸(x¸, y¸, z¸) a P(x, y, z)

L

r¸ O

r

v

x y 1

FIGURE 1

r 苷 r0 ⫹ t v

SECTION 12.5 EQUATIONS OF LINES AND PLANES

z

L

t0

t=0

||||

y

具x, y, z 典 苷 具x 0 ⫹ ta, y0 ⫹ tb, z0 ⫹ tc 典

FIGURE 2

Two vectors are equal if and only if corresponding components are equal. Therefore we have the three scalar equations:

2

x 苷 x 0 ⫹ at

y 苷 y0 ⫹ bt

z 苷 z0 ⫹ ct

where t 僆 ⺢. These equations are called parametric equations of the line L through the point P0共x 0 , y0 , z0兲 and parallel to the vector v 苷 具a, b, c典 . Each value of the parameter t gives a point 共x, y, z兲 on L. Figure 3 shows the line L in Example 1 and its relation to the given point and to the vector that gives its direction.

N

z

(a) Here r0 苷 具 5, 1, 3典 苷 5i ⫹ j ⫹ 3k and v 苷 i ⫹ 4 j ⫺ 2k, so the vector equation (1) becomes

r¸ v=i+4j-2k

x

(a) Find a vector equation and parametric equations for the line that passes through the point 共5, 1, 3兲 and is parallel to the vector i ⫹ 4 j ⫺ 2 k. (b) Find two other points on the line. SOLUTION

L (5, 1, 3)

EXAMPLE 1

r 苷 共5i ⫹ j ⫹ 3k兲 ⫹ t共i ⫹ 4 j ⫺ 2k兲

y

or

r 苷 共5 ⫹ t兲 i ⫹ 共1 ⫹ 4t兲 j ⫹ 共3 ⫺ 2t兲 k

Parametric equations are FIGURE 3

x苷5⫹t

y 苷 1 ⫹ 4t

z 苷 3 ⫺ 2t

(b) Choosing the parameter value t 苷 1 gives x 苷 6, y 苷 5, and z 苷 1, so 共6, 5, 1兲 is a M point on the line. Similarly, t 苷 ⫺1 gives the point 共4, ⫺3, 5兲. The vector equation and parametric equations of a line are not unique. If we change the point or the parameter or choose a different parallel vector, then the equations change. For instance, if, instead of 共5, 1, 3兲, we choose the point 共6, 5, 1兲 in Example 1, then the parametric equations of the line become x苷6⫹t

y 苷 5 ⫹ 4t

z 苷 1 ⫺ 2t

Or, if we stay with the point 共5, 1, 3兲 but choose the parallel vector 2i ⫹ 8 j ⫺ 4k, we arrive at the equations x 苷 5 ⫹ 2t

y 苷 1 ⫹ 8t

z 苷 3 ⫺ 4t

In general, if a vector v 苷 具 a, b, c典 is used to describe the direction of a line L, then the numbers a, b, and c are called direction numbers of L. Since any vector parallel to v

796

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CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

could also be used, we see that any three numbers proportional to a, b, and c could also be used as a set of direction numbers for L. Another way of describing a line L is to eliminate the parameter t from Equations 2. If none of a, b, or c is 0, we can solve each of these equations for t, equate the results, and obtain

3

x ⫺ x0 y ⫺ y0 z ⫺ z0 苷 苷 a b c

These equations are called symmetric equations of L . Notice that the numbers a, b, and c that appear in the denominators of Equations 3 are direction numbers of L, that is, components of a vector parallel to L. If one of a, b, or c is 0, we can still eliminate t. For instance, if a 苷 0, we could write the equations of L as x 苷 x0

y ⫺ y0 z ⫺ z0 苷 b c

This means that L lies in the vertical plane x 苷 x 0. EXAMPLE 2

z

(a) Find parametric equations and symmetric equations of the line that passes through the points A共2, 4, ⫺3兲 and B共3, ⫺1, 1兲. (b) At what point does this line intersect the xy-plane?

1

SOLUTION

Figure 4 shows the line L in Example 2 and the point P where it intersects the xy-plane.

N

B x

2

1

P

_1

y

v 苷 具 3 ⫺ 2, ⫺1 ⫺ 4, 1 ⫺ 共⫺3兲典 苷 具1, ⫺5, 4典

L

A FIGURE 4

(a) We are not explicitly given a vector parallel to the line, but observe that the vector v l with representation AB is parallel to the line and

4

Thus direction numbers are a 苷 1, b 苷 ⫺5, and c 苷 4. Taking the point 共2, 4, ⫺3兲 as P0, we see that parametric equations (2) are x苷2⫹t

y 苷 4 ⫺ 5t

z 苷 ⫺3 ⫹ 4t

and symmetric equations (3) are x⫺2 y⫺4 z⫹3 苷 苷 1 ⫺5 4 (b) The line intersects the xy-plane when z 苷 0, so we put z 苷 0 in the symmetric equations and obtain x⫺2 y⫺4 3 苷 苷 1 ⫺5 4 This gives x 苷 114 and y 苷 14 , so the line intersects the xy-plane at the point ( 114 , 14 , 0).

M

In general, the procedure of Example 2 shows that direction numbers of the line L through the points P0共x 0 , y0 , z0 兲 and P1共x 1, y1, z1兲 are x 1 ⫺ x 0 , y1 ⫺ y0 , and z1 ⫺ z0 and so symmetric equations of L are x ⫺ x0 y ⫺ y0 z ⫺ z0 苷 苷 x1 ⫺ x0 y1 ⫺ y0 z1 ⫺ z0

SECTION 12.5 EQUATIONS OF LINES AND PLANES

||||

797

Often, we need a description, not of an entire line, but of just a line segment. How, for instance, could we describe the line segment AB in Example 2? If we put t 苷 0 in the parametric equations in Example 2(a), we get the point 共2, 4, ⫺3兲 and if we put t 苷 1 we get 共3, ⫺1, 1兲. So the line segment AB is described by the parametric equations x苷2⫹t

y 苷 4 ⫺ 5t

z 苷 ⫺3 ⫹ 4t

0艋t艋1

or by the corresponding vector equation r共t兲 苷 具2 ⫹ t, 4 ⫺ 5t, ⫺3 ⫹ 4t典

0艋t艋1

In general, we know from Equation 1 that the vector equation of a line through the (tip of the) vector r 0 in the direction of a vector v is r 苷 r 0 ⫹ t v. If the line also passes through (the tip of) r1, then we can take v 苷 r1 ⫺ r 0 and so its vector equation is r 苷 r 0 ⫹ t 共r1 ⫺ r 0兲 苷 共1 ⫺ t兲r 0 ⫹ t r1 The line segment from r 0 to r1 is given by the parameter interval 0 艋 t 艋 1. 4

The line segment from r 0 to r1 is given by the vector equation r共t兲 苷 共1 ⫺ t兲r 0 ⫹ t r1

The lines L 1 and L 2 in Example 3, shown in Figure 5, are skew lines.

N

V EXAMPLE 3

z

L¡

5

0艋t艋1

Show that the lines L 1 and L 2 with parametric equations x苷1⫹t

y 苷 ⫺2 ⫹ 3t

z苷4⫺t

x 苷 2s

y苷3⫹s

z 苷 ⫺3 ⫹ 4s

are skew lines; that is, they do not intersect and are not parallel (and therefore do not lie in the same plane).

L™

SOLUTION The lines are not parallel because the corresponding vectors 具 1, 3, ⫺1典 and

5 10

5 x

y

具2, 1, 4典 are not parallel. (Their components are not proportional.) If L 1 and L 2 had a point of intersection, there would be values of t and s such that 1 ⫹ t 苷 2s

_5

⫺2 ⫹ 3t 苷 3 ⫹ s 4 ⫺ t 苷 ⫺3 ⫹ 4s

FIGURE 5

But if we solve the first two equations, we get t 苷 115 and s 苷 85 , and these values don’t satisfy the third equation. Therefore there are no values of t and s that satisfy the three equations, so L 1 and L 2 do not intersect. Thus L 1 and L 2 are skew lines.

z

n

PLANES

P(x, y, z)

r 0

r-r¸ r¸ P¸(x¸, y¸, z¸)

x y

FIGURE 6

M

Although a line in space is determined by a point and a direction, a plane in space is more difficult to describe. A single vector parallel to a plane is not enough to convey the “direction” of the plane, but a vector perpendicular to the plane does completely specify its direction. Thus a plane in space is determined by a point P0共x 0 , y0 , z0兲 in the plane and a vector n that is orthogonal to the plane. This orthogonal vector n is called a normal vector. Let P共x, y, z兲 be an arbitrary point in the plane, and let r0 and r be the position vectors of P0 and P. Then the vector r ⫺ r0 is represented by P A. 0 P (See Figure 6.) The normal vector n is orthogonal to every vector in the given plane. In particular, n is orthogonal

798

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CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

to r ⫺ r0 and so we have n ⴢ 共r ⫺ r0 兲 苷 0

5

which can be rewritten as n ⴢ r 苷 n ⴢ r0

6

Either Equation 5 or Equation 6 is called a vector equation of the plane. To obtain a scalar equation for the plane, we write n 苷 具a, b, c 典 , r 苷 具x, y, z典 , and r0 苷 具x 0 , y0 , z0 典 . Then the vector equation (5) becomes 具a, b, c典 ⴢ 具x ⫺ x 0 , y ⫺ y0 , z ⫺ z0 典 苷 0 or 7

a共x ⫺ x 0 兲 ⫹ b共y ⫺ y0 兲 ⫹ c共z ⫺ z0 兲 苷 0

Equation 7 is the scalar equation of the plane through P0共x 0 , y0 , z0 兲 with normal vector n 苷 具a, b, c 典 . Find an equation of the plane through the point 共2, 4, ⫺1兲 with normal vector n 苷 具2, 3, 4典 . Find the intercepts and sketch the plane. V EXAMPLE 4

SOLUTION Putting a 苷 2, b 苷 3, c 苷 4, x 0 苷 2, y0 苷 4, and z0 苷 ⫺1 in Equation 7, we

z

see that an equation of the plane is

(0, 0, 3)

2共x ⫺ 2兲 ⫹ 3共 y ⫺ 4兲 ⫹ 4共z ⫹ 1兲 苷 0 (0, 4, 0) (6, 0, 0) x

FIGURE 7

2x ⫹ 3y ⫹ 4z 苷 12

or y

To find the x-intercept we set y 苷 z 苷 0 in this equation and obtain x 苷 6. Similarly, the y-intercept is 4 and the z-intercept is 3. This enables us to sketch the portion of the plane that lies in the first octant (see Figure 7).

M

By collecting terms in Equation 7 as we did in Example 4, we can rewrite the equation of a plane as ax ⫹ by ⫹ cz ⫹ d 苷 0

8

where d 苷 ⫺共ax 0 ⫹ by0 ⫹ cz0 兲. Equation 8 is called a linear equation in x, y, and z. Conversely, it can be shown that if a, b, and c are not all 0, then the linear equation (8) represents a plane with normal vector 具a, b, c典 . (See Exercise 77.) EXAMPLE 5 Find an equation of the plane that passes through the points P共1, 3, 2兲,

Q共3, ⫺1, 6兲, and R共5, 2, 0兲. l

l

SOLUTION The vectors a and b corresponding to PQ and PR are

a 苷 具2, ⫺4, 4典

b 苷 具4, ⫺1, ⫺2典

SECTION 12.5 EQUATIONS OF LINES AND PLANES

Figure 8 shows the portion of the plane in Example 5 that is enclosed by triangle PQR.

N

||||

799

Since both a and b lie in the plane, their cross product a ⫻ b is orthogonal to the plane and can be taken as the normal vector. Thus

ⱍ ⱍ

z

i n苷a⫻b苷 2 4

Q(3, _1, 6)

P(1, 3, 2)

j ⫺4 ⫺1

k 4 苷 12 i ⫹ 20 j ⫹ 14 k ⫺2

With the point P共1, 3, 2兲 and the normal vector n, an equation of the plane is y

12共x ⫺ 1兲 ⫹ 20共y ⫺ 3兲 ⫹ 14共z ⫺ 2兲 苷 0

x

R(5, 2, 0)

6x ⫹ 10y ⫹ 7z 苷 50

or

M

FIGURE 8

EXAMPLE 6 Find the point at which the line with parametric equations x 苷 2 ⫹ 3t,

y 苷 ⫺4t, z 苷 5 ⫹ t intersects the plane 4x ⫹ 5y ⫺ 2z 苷 18.

SOLUTION We substitute the expressions for x, y, and z from the parametric equations into

the equation of the plane: 4共2 ⫹ 3t兲 ⫹ 5共⫺4t兲 ⫺ 2共5 ⫹ t兲 苷 18 This simplifies to ⫺10t 苷 20, so t 苷 ⫺2. Therefore the point of intersection occurs when the parameter value is t 苷 ⫺2. Then x 苷 2 ⫹ 3共⫺2兲 苷 ⫺4, y 苷 ⫺4共⫺2兲 苷 8, z 苷 5 ⫺ 2 苷 3 and so the point of intersection is 共⫺4, 8, 3兲. n™ ¨ n¡

Two planes are parallel if their normal vectors are parallel. For instance, the planes x ⫹ 2y ⫺ 3z 苷 4 and 2x ⫹ 4y ⫺ 6z 苷 3 are parallel because their normal vectors are n1 苷 具1, 2, ⫺3典 and n 2 苷 具 2, 4, ⫺6典 and n 2 苷 2n1 . If two planes are not parallel, then they intersect in a straight line and the angle between the two planes is defined as the acute angle between their normal vectors (see angle ␪ in Figure 9).

¨ FIGURE 9 Figure 10 shows the planes in Example 7 and their line of intersection L.

N

x-2y+3z=1

x+y+z=1

M

V EXAMPLE 7

(a) Find the angle between the planes x ⫹ y ⫹ z 苷 1 and x ⫺ 2y ⫹ 3z 苷 1. (b) Find symmetric equations for the line of intersection L of these two planes. SOLUTION

(a) The normal vectors of these planes are

6 4 2 z 0 _2 _4

L

n1 苷 具 1, 1, 1典

n 2 苷 具 1, ⫺2, 3典

and so, if ␪ is the angle between the planes, Corollary 12.3.6 gives _2

0 y

FIGURE 10

2

2

0 x

_2

cos ␪ 苷

n1 ⴢ n 2 1共1兲 ⫹ 1共⫺2兲 ⫹ 1共3兲 2 苷 苷 n1 n 2 s1 ⫹ 1 ⫹ 1 s1 ⫹ 4 ⫹ 9 s42

ⱍ ⱍⱍ ⱍ

冉 冊

␪ 苷 cos⫺1

2 s42

⬇ 72⬚

(b) We first need to find a point on L. For instance, we can find the point where the line intersects the xy-plane by setting z 苷 0 in the equations of both planes. This gives the equations x ⫹ y 苷 1 and x ⫺ 2y 苷 1, whose solution is x 苷 1, y 苷 0. So the point 共1, 0, 0兲 lies on L.

800

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CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

Now we observe that, since L lies in both planes, it is perpendicular to both of the normal vectors. Thus a vector v parallel to L is given by the cross product Another way to find the line of intersection is to solve the equations of the planes for two of the variables in terms of the third, which can be taken as the parameter.

N

v 苷 n1 ⫻ n 2 苷

ⱍ ⱍ i j 1 1 1 ⫺2

k 1 苷 5i ⫺ 2 j ⫺ 3 k 3

and so the symmetric equations of L can be written as x⫺1 y z 苷 苷 5 ⫺2 ⫺3

y x-1 = _2 5

2

L

1 z 0 y _1

2

z

=3

NOTE Since a linear equation in x, y, and z represents a plane and two nonparallel planes intersect in a line, it follows that two linear equations can represent a line. The points 共x, y, z兲 that satisfy both a 1 x ⫹ b1 y ⫹ c1 z ⫹ d1 苷 0 and a 2 x ⫹ b2 y ⫹ c2 z ⫹ d2 苷 0 lie on both of these planes, and so the pair of linear equations represents the line of intersection of the planes (if they are not parallel). For instance, in Example 7 the line L was given as the line of intersection of the planes x ⫹ y ⫹ z 苷 1 and x ⫺ 2y ⫹ 3z 苷 1. The symmetric equations that we found for L could be written as

x⫺1 y 苷 5 ⫺2

_2 _1 y

0

1

_2 0 _1 x

1

2

FIGURE 11 Figure 11 shows how the line L in Example 7 can also be regarded as the line of intersection of planes derived from its symmetric equations.

M

and

y z 苷 ⫺2 ⫺3

which is again a pair of linear equations. They exhibit L as the line of intersection of the planes 共x ⫺ 1兲兾5 苷 y兾共⫺2兲 and y兾共⫺2兲 苷 z兾共⫺3兲. (See Figure 11.) In general, when we write the equations of a line in the symmetric form

N

x ⫺ x0 y ⫺ y0 z ⫺ z0 苷 苷 a b c we can regard the line as the line of intersection of the two planes x ⫺ x0 y ⫺ y0 苷 a b

and

y ⫺ y0 z ⫺ z0 苷 b c

EXAMPLE 8 Find a formula for the distance D from a point P1共x 1, y1, z1兲 to the

plane ax ⫹ by ⫹ cz ⫹ d 苷 0.

SOLUTION Let P0共x 0 , y0 , z0 兲 be any point in the given plane and let b be the vector

corresponding to PA. 0 P1 Then b 苷 具x 1 ⫺ x 0 , y1 ⫺ y0 , z1 ⫺ z0 典 P¡

From Figure 12 you can see that the distance D from P1 to the plane is equal to the absolute value of the scalar projection of b onto the normal vector n 苷 具 a, b, c典 . (See Section 12.3.) Thus

¨ b

P¸

D n

ⱍ

nⴢb ⱍ ⱍ ⱍnⱍ ⱍ

D 苷 compn b 苷 苷

ⱍ a共x

⫺ x0 兲 ⫹ b共y1 ⫺ y0 兲 ⫹ c共z1 ⫺ z0 兲 sa 2 ⫹ b 2 ⫹ c 2

苷

ⱍ 共ax

⫹ by1 ⫹ cz1 兲 ⫺ 共ax0 ⫹ by0 ⫹ cz0 兲 sa 2 ⫹ b 2 ⫹ c 2

FIGURE 12

1

1

ⱍ ⱍ

SECTION 12.5 EQUATIONS OF LINES AND PLANES

||||

801

Since P0 lies in the plane, its coordinates satisfy the equation of the plane and so we have ax 0 ⫹ by0 ⫹ cz0 ⫹ d 苷 0. Thus the formula for D can be written as

D苷

9

ⱍ ax

⫹ by1 ⫹ cz1 ⫹ d sa 2 ⫹ b 2 ⫹ c 2

1

ⱍ

M

EXAMPLE 9 Find the distance between the parallel planes 10x ⫹ 2y ⫺ 2z 苷 5

and 5x ⫹ y ⫺ z 苷 1.

SOLUTION First we note that the planes are parallel because their normal vectors

具 10, 2, ⫺2典 and 具5, 1, ⫺1典 are parallel. To find the distance D between the planes, we choose any point on one plane and calculate its distance to the other plane. In particular, if we put y 苷 z 苷 0 in the equation of the first plane, we get 10x 苷 5 and so ( 12 , 0, 0) is a point in this plane. By Formula 9, the distance between ( 12 , 0, 0) and the plane 5x ⫹ y ⫺ z ⫺ 1 苷 0 is D苷

ⱍ 5( ) ⫹ 1共0兲 ⫺ 1共0兲 ⫺ 1 ⱍ 苷 1 2

s5 2 ⫹ 12 ⫹ 共⫺1兲2

3 2

3s3

苷

s3 6

So the distance between the planes is s3 兾6.

M

EXAMPLE 10 In Example 3 we showed that the lines

L1:

x苷1⫹t

y 苷 ⫺2 ⫹ 3t

z苷4⫺t

L2:

x 苷 2s

y苷3⫹s

z 苷 ⫺3 ⫹ 4s

are skew. Find the distance between them. SOLUTION Since the two lines L 1 and L 2 are skew, they can be viewed as lying on two

parallel planes P1 and P2 . The distance between L 1 and L 2 is the same as the distance between P1 and P2 , which can be computed as in Example 9. The common normal vector to both planes must be orthogonal to both v1 苷 具 1, 3, ⫺1典 (the direction of L 1 ) and v2 苷 具 2, 1, 4典 (the direction of L 2 ). So a normal vector is n 苷 v1 ⫻ v2 苷

ⱍ ⱍ i j 1 3 2 1

k ⫺1 苷 13i ⫺ 6 j ⫺ 5k 4

If we put s 苷 0 in the equations of L 2 , we get the point 共0, 3, ⫺3兲 on L 2 and so an equation for P2 is 13共x ⫺ 0兲 ⫺ 6共y ⫺ 3兲 ⫺ 5共z ⫹ 3兲 苷 0

or

13x ⫺ 6y ⫺ 5z ⫹ 3 苷 0

If we now set t 苷 0 in the equations for L 1 , we get the point 共1, ⫺2, 4兲 on P1 . So the distance between L 1 and L 2 is the same as the distance from 共1, ⫺2, 4兲 to 13x ⫺ 6y ⫺ 5z ⫹ 3 苷 0. By Formula 9, this distance is D苷

ⱍ 13共1兲 ⫺ 6共⫺2兲 ⫺ 5共4兲 ⫹ 3 ⱍ 苷 s13 ⫹ 共⫺6兲 ⫹ 共⫺5兲 2

2

2

8 ⬇ 0.53 s230

M

802

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12.5

CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

EXERCISES

1. Determine whether each statement is true or false.

(a) (b) (c) (d) (e) (f) (g) (h) (i) ( j) (k)

Two lines parallel to a third line are parallel. Two lines perpendicular to a third line are parallel. Two planes parallel to a third plane are parallel. Two planes perpendicular to a third plane are parallel. Two lines parallel to a plane are parallel. Two lines perpendicular to a plane are parallel. Two planes parallel to a line are parallel. Two planes perpendicular to a line are parallel. Two planes either intersect or are parallel. Two lines either intersect or are parallel. A plane and a line either intersect or are parallel.

2–5 Find a vector equation and parametric equations for the line. 2. The line through the point 共6, ⫺5, 2兲 and parallel to the

vector 具 1, 3, ⫺ 23 典

3. The line through the point 共2, 2.4, 3.5兲 and parallel to the

vector 3 i ⫹ 2 j ⫺ k

4. The line through the point 共0, 14, ⫺10兲 and parallel to the line

x 苷 ⫺1 ⫹ 2t, y 苷 6 ⫺ 3t, z 苷 3 ⫹ 9t

5. The line through the point (1, 0, 6) and perpendicular to the

plane x ⫹ 3y ⫹ z 苷 5 6 –12 Find parametric equations and symmetric equations for the

16. (a) Find parametric equations for the line through 共2, 4, 6兲 that

is perpendicular to the plane x ⫺ y ⫹ 3z 苷 7. (b) In what points does this line intersect the coordinate planes?

17. Find a vector equation for the line segment from 共2, ⫺1, 4兲

to 共4, 6, 1兲.

18. Find parametric equations for the line segment from 共10, 3, 1兲

to 共5, 6, ⫺3兲.

19–22 Determine whether the lines L 1 and L 2 are parallel, skew, or

intersecting. If they intersect, find the point of intersection. 19. L 1: x 苷 ⫺6t,

y 苷 1 ⫹ 9t,

L 2: x 苷 1 ⫹ 2s, 20. L 1: x 苷 1 ⫹ 2t,

L 2: x 苷 ⫺1 ⫹ s, 21. L 1:

z 苷 ⫺3t

y 苷 4 ⫺ 3s, y 苷 3t,

z苷s

z苷2⫺t

y 苷 4 ⫹ s,

z 苷 1 ⫹ 3s

y⫺1 z⫺2 x 苷 苷 1 2 3

L 2:

y⫺2 z⫺1 x⫺3 苷 苷 ⫺4 ⫺3 2

22. L 1:

x⫺1 y⫺3 z⫺2 苷 苷 2 2 ⫺1

L 2:

x⫺2 y⫺6 z⫹2 苷 苷 1 ⫺1 3

line. 6. The line through the origin and the point 共1, 2, 3兲

23–38 Find an equation of the plane.

7. The line through the points 共1, 3, 2兲 and 共⫺4, 3, 0兲

23. The plane through the point 共6, 3, 2兲 and perpendicular to the

8. The line through the points 共6, 1, ⫺3兲 and 共2, 4, 5兲 9. The line through the points (0, 12 , 1) and 共2, 1, ⫺3兲 10. The line through 共2, 1, 0兲 and perpendicular to both i ⫹ j

and j ⫹ k

11. The line through 共1, ⫺1, 1兲 and parallel to the line 1 x ⫹ 2 苷 2y 苷 z ⫺ 3

12. The line of intersection of the planes x ⫹ y ⫹ z 苷 1

and x ⫹ z 苷 0

13. Is the line through 共⫺4, ⫺6, 1兲 and 共⫺2, 0 ⫺3兲 parallel to the

line through 共10, 18, 4兲 and 共5, 3, 14兲 ?

14. Is the line through 共4, 1, ⫺1兲 and 共2, 5, 3兲 perpendicular to the

line through 共⫺3, 2, 0兲 and 共5, 1, 4兲 ?

15. (a) Find symmetric equations for the line that passes through

the point 共1, ⫺5, 6兲 and is parallel to the vector 具⫺1, 2, ⫺3 典 . (b) Find the points in which the required line in part (a) intersects the coordinate planes.

vector 具⫺2, 1, 5典

24. The plane through the point 共4, 0, ⫺3兲 and with normal

vector j ⫹ 2 k

25. The plane through the point 共1, ⫺1, 1兲 and with normal

vector i ⫹ j ⫺ k

26. The plane through the point 共⫺2, 8, 10兲 and perpendicular to

the line x 苷 1 ⫹ t, y 苷 2t, z 苷 4 ⫺ 3t

27. The plane through the origin and parallel to the plane

2 x ⫺ y ⫹ 3z 苷 1 28. The plane through the point 共⫺1, 6, ⫺5兲 and parallel to the

plane x ⫹ y ⫹ z ⫹ 2 苷 0

29. The plane through the point 共4, ⫺2, 3兲 and parallel to the plane

3x ⫺ 7z 苷 12

30. The plane that contains the line x 苷 3 ⫹ 2t, y 苷 t, z 苷 8 ⫺ t

and is parallel to the plane 2 x ⫹ 4y ⫹ 8z 苷 17

31. The plane through the points 共0, 1, 1兲, 共1, 0, 1兲, and 共1, 1, 0兲 32. The plane through the origin and the points 共2, ⫺4, 6兲

and 共5, 1, 3兲

SECTION 12.5 EQUATIONS OF LINES AND PLANES

33. The plane through the points 共3, ⫺1, 2兲, 共8, 2, 4兲, and

共⫺1, ⫺2, ⫺3兲

||||

803

57–58 Find symmetric equations for the line of intersection of the

planes.

34. The plane that passes through the point 共1, 2, 3兲 and contains

the line x 苷 3t, y 苷 1 ⫹ t, z 苷 2 ⫺ t

35. The plane that passes through the point 共6, 0, ⫺2兲 and contains

the line x 苷 4 ⫺ 2t, y 苷 3 ⫹ 5t, z 苷 7 ⫹ 4t

36. The plane that passes through the point 共1, ⫺1, 1兲 and

contains the line with symmetric equations x 苷 2y 苷 3z

37. The plane that passes through the point 共⫺1, 2, 1兲 and contains

the line of intersection of the planes x ⫹ y ⫺ z 苷 2 and 2 x ⫺ y ⫹ 3z 苷 1

38. The plane that passes through the line of intersection of the

planes x ⫺ z 苷 1 and y ⫹ 2z 苷 3 and is perpendicular to the plane x ⫹ y ⫺ 2z 苷 1

57. 5x ⫺ 2y ⫺ 2z 苷 1, 58. z 苷 2x ⫺ y ⫺ 5,

4x ⫹ y ⫹ z 苷 6

z 苷 4x ⫹ 3y ⫺ 5

59. Find an equation for the plane consisting of all points that are

equidistant from the points 共1, 0, ⫺2兲 and 共3, 4, 0兲. 60. Find an equation for the plane consisting of all points that are

equidistant from the points 共2, 5, 5兲 and 共⫺6, 3, 1兲. 61. Find an equation of the plane with x-intercept a, y-intercept b,

and z-intercept c. 62. (a) Find the point at which the given lines intersect:

r 苷 具1, 1, 0 典 ⫹ t 具1, ⫺1, 2典 r 苷 具2, 0, 2典 ⫹ s具 ⫺1, 1, 0典

39– 42 Use intercepts to help sketch the plane. 39. 2x ⫹ 5y ⫹ z 苷 10

40. 3x ⫹ y ⫹ 2z 苷 6

41. 6x ⫺ 3y ⫹ 4z 苷 6

42. 6x ⫹ 5y ⫺ 3z 苷 15

43– 45 Find the point at which the line intersects the given plane. 43. x 苷 3 ⫺ t, y 苷 2 ⫹ t, z 苷 5t ;

x ⫺ y ⫹ 2z 苷 9

44. x 苷 1 ⫹ 2t, y 苷 4t, z 苷 2 ⫺ 3t ; 45. x 苷 y ⫺ 1 苷 2z ;

x ⫹ 2y ⫺ z ⫹ 1 苷 0

4x ⫺ y ⫹ 3z 苷 8

46. Where does the line through 共1, 0, 1兲 and 共4, ⫺2, 2兲 intersect

the plane x ⫹ y ⫹ z 苷 6 ?

47. Find direction numbers for the line of intersection of the planes

x ⫹ y ⫹ z 苷 1 and x ⫹ z 苷 0.

(b) Find an equation of the plane that contains these lines. 63. Find parametric equations for the line through the point

共0, 1, 2兲 that is parallel to the plane x ⫹ y ⫹ z 苷 2 and perpendicular to the line x 苷 1 ⫹ t, y 苷 1 ⫺ t, z 苷 2t. 64. Find parametric equations for the line through the point

共0, 1, 2兲 that is perpendicular to the line x 苷 1 ⫹ t, y 苷 1 ⫺ t, z 苷 2t and intersects this line. 65. Which of the following four planes are parallel? Are any of

them identical? P1 : 4x ⫺ 2y ⫹ 6z 苷 3

P2 : 4x ⫺ 2y ⫺ 2z 苷 6

P3 : ⫺6x ⫹ 3y ⫺ 9z 苷 5

P4 : z 苷 2 x ⫺ y ⫺ 3

66. Which of the following four lines are parallel? Are any of them

identical?

48. Find the cosine of the angle between the planes x ⫹ y ⫹ z 苷 0

L 1 : x 苷 1 ⫹ t,

and x ⫹ 2y ⫹ 3z 苷 1.

L 3 : x 苷 1 ⫹ t,

neither. If neither, find the angle between them. 50. 2z 苷 4y ⫺ x,

x⫺y⫹z苷1

52. 2 x ⫺ 3y ⫹ 4z 苷 5 , 53. x 苷 4y ⫺ 2z,

x ⫹ 6y ⫹ 4z 苷 3

8y 苷 1 ⫹ 2 x ⫹ 4z

54. x ⫹ 2y ⫹ 2z 苷 1,

2x ⫺ y ⫹ 2z 苷 1

55–56 (a) Find parametric equations for the line of intersection of

the planes and (b) find the angle between the planes. 55. x ⫹ y ⫹ z 苷 1,

x ⫹ 2y ⫹ 2z 苷 1

56. 3x ⫺ 2y ⫹ z 苷 1,

2x ⫹ y ⫺ 3z 苷 3

y 苷 4 ⫹ t,

z苷1⫺t

L 4 : r 苷 具2, 1, ⫺3典 ⫹ t 具2, 2, ⫺10典

⫺3x ⫹ 6y ⫹ 7z 苷 0

3x ⫺ 12y ⫹ 6z 苷 1

51. x ⫹ y ⫹ z 苷 1,

z 苷 2 ⫺ 5t

L2: x ⫹ 1 苷 y ⫺ 2 苷 1 ⫺ z

49–54 Determine whether the planes are parallel, perpendicular, or 49. x ⫹ 4y ⫺ 3z 苷 1,

y 苷 t,

67–68 Use the formula in Exercise 43 in Section 12.4 to find the

distance from the point to the given line. 67. 共4, 1, ⫺2兲; 68. 共0, 1, 3兲;

x 苷 1 ⫹ t, y 苷 3 ⫺ 2t, z 苷 4 ⫺ 3t x 苷 2t, y 苷 6 ⫺ 2t, z 苷 3 ⫹ t

69–70 Find the distance from the point to the given plane. 69. 共1, ⫺2, 4兲,

3x ⫹ 2y ⫹ 6z 苷 5

70. 共⫺6, 3, 5兲,

x ⫺ 2y ⫺ 4z 苷 8

71–72 Find the distance between the given parallel planes. 71. 2x ⫺ 3y ⫹ z 苷 4,

4x ⫺ 6y ⫹ 2z 苷 3

804

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CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

72. 6z 苷 4y ⫺ 2x,

9z 苷 1 ⫺ 3x ⫹ 6y

76. Find the distance between the skew lines with para-

73. Show that the distance between the parallel planes

ax ⫹ by ⫹ cz ⫹ d1 苷 0 and ax ⫹ by ⫹ cz ⫹ d2 苷 0 is

ⱍd

ⱍ

⫺ d2 D苷 sa 2 ⫹ b 2 ⫹ c 2 1

metric equations x 苷 1 ⫹ t, y 苷 1 ⫹ 6t, z 苷 2t, and x 苷 1 ⫹ 2s, y 苷 5 ⫹ 15s, z 苷 ⫺2 ⫹ 6s. 77. If a, b, and c are not all 0, show that the equation

ax ⫹ by ⫹ cz ⫹ d 苷 0 represents a plane and 具a, b, c典 is a normal vector to the plane. Hint: Suppose a 苷 0 and rewrite the equation in the form

冉 冊

74. Find equations of the planes that are parallel to the plane

a x⫹

x ⫹ 2y ⫺ 2z 苷 1 and two units away from it. 75. Show that the lines with symmetric equations x 苷 y 苷 z and

x ⫹ 1 苷 y兾2 苷 z兾3 are skew, and find the distance between these lines.

L A B O R AT O R Y PROJECT

d a

⫹ b共 y ⫺ 0兲 ⫹ c共z ⫺ 0兲 苷 0

78. Give a geometric description of each family of planes.

(a) x ⫹ y ⫹ z 苷 c (c) y cos ␪ ⫹ z sin ␪ 苷 1

(b) x ⫹ y ⫹ cz 苷 1

PUTTING 3D IN PERSPECTIVE Computer graphics programmers face the same challenge as the great painters of the past: how to represent a three-dimensional scene as a flat image on a two-dimensional plane (a screen or a canvas). To create the illusion of perspective, in which closer objects appear larger than those farther away, three-dimensional objects in the computer’s memory are projected onto a rectangular screen window from a viewpoint where the eye, or camera, is located. The viewing volume––the portion of space that will be visible––is the region contained by the four planes that pass through the viewpoint and an edge of the screen window. If objects in the scene extend beyond these four planes, they must be truncated before pixel data are sent to the screen. These planes are therefore called clipping planes. 1. Suppose the screen is represented by a rectangle in the yz-plane with vertices 共0, ⫾400, 0兲

and 共0, ⫾400, 600兲, and the camera is placed at 共1000, 0, 0兲. A line L in the scene passes through the points 共230, ⫺285, 102兲 and 共860, 105, 264兲. At what points should L be clipped by the clipping planes?

2. If the clipped line segment is projected on the screen window, identify the resulting line

segment. 3. Use parametric equations to plot the edges of the screen window, the clipped line segment,

and its projection on the screen window. Then add sight lines connecting the viewpoint to each end of the clipped segments to verify that the projection is correct. 4. A rectangle with vertices 共621, ⫺147, 206兲, 共563, 31, 242兲, 共657, ⫺111, 86兲, and

共599, 67, 122兲 is added to the scene. The line L intersects this rectangle. To make the rectangle appear opaque, a programmer can use hidden line rendering, which removes portions of objects that are behind other objects. Identify the portion of L that should be removed.

12.6 CYLINDERS AND QUADRIC SURFACES We have already looked at two special types of surfaces: planes (in Section 12.5) and spheres (in Section 12.1). Here we investigate two other types of surfaces: cylinders and quadric surfaces. In order to sketch the graph of a surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes. These curves are called traces (or cross-sections) of the surface.

SECTION 12.6 CYLINDERS AND QUADRIC SURFACES

||||

805

CYLINDERS

A cylinder is a surface that consists of all lines (called rulings) that are parallel to a given line and pass through a given plane curve. z

V EXAMPLE 1

Sketch the graph of the surface z 苷 x 2.

SOLUTION Notice that the equation of the graph, z 苷 x 2, doesn’t involve y. This means that

any vertical plane with equation y 苷 k (parallel to the xz-plane) intersects the graph in a curve with equation z 苷 x 2. So these vertical traces are parabolas. Figure 1 shows how the graph is formed by taking the parabola z 苷 x 2 in the xz-plane and moving it in the direction of the y-axis. The graph is a surface, called a parabolic cylinder, made up of infinitely many shifted copies of the same parabola. Here the rulings of the cylinder are parallel to the y-axis. M

0 x

y

We noticed that the variable y is missing from the equation of the cylinder in Example 1. This is typical of a surface whose rulings are parallel to one of the coordinate axes. If one of the variables x, y, or z is missing from the equation of a surface, then the surface is a cylinder.

FIGURE 1

The surface z=≈ is a parabolic cylinder.

EXAMPLE 2 Identify and sketch the surfaces.

(a) x 2 ⫹ y 2 苷 1

(b) y 2 ⫹ z 2 苷 1

SOLUTION

(a) Since z is missing and the equations x 2 ⫹ y 2 苷 1, z 苷 k represent a circle with radius 1 in the plane z 苷 k, the surface x 2 ⫹ y 2 苷 1 is a circular cylinder whose axis is the z-axis. (See Figure 2.) Here the rulings are vertical lines. (b) In this case x is missing and the surface is a circular cylinder whose axis is the x-axis. (See Figure 3.) It is obtained by taking the circle y 2 ⫹ z 2 苷 1, x 苷 0 in the yz-plane and moving it parallel to the x-axis. z

z

y 0

x y

x

FIGURE 2 ≈+¥=1

|

FIGURE 3 ¥+z@=1

M

NOTE When you are dealing with surfaces, it is important to recognize that an equation like x 2 ⫹ y 2 苷 1 represents a cylinder and not a circle. The trace of the cylinder x 2 ⫹ y 2 苷 1 in the xy-plane is the circle with equations x 2 ⫹ y 2 苷 1, z 苷 0. QUADRIC SURFACES

A quadric surface is the graph of a second-degree equation in three variables x, y, and z. The most general such equation is Ax 2 ⫹ By 2 ⫹ Cz 2 ⫹ Dxy ⫹ Eyz ⫹ Fxz ⫹ Gx ⫹ Hy ⫹ Iz ⫹ J 苷 0

806

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CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

where A, B, C, . . . , J are constants, but by translation and rotation it can be brought into one of the two standard forms Ax 2 ⫹ By 2 ⫹ Cz 2 ⫹ J 苷 0

or

Ax 2 ⫹ By 2 ⫹ Iz 苷 0

Quadric surfaces are the counterparts in three dimensions of the conic sections in the plane. (See Section 10.5 for a review of conic sections.) EXAMPLE 3 Use traces to sketch the quadric surface with equation

x2 ⫹

y2 z2 ⫹ 苷1 9 4

SOLUTION By substituting z 苷 0, we find that the trace in the xy-plane is x 2 ⫹ y 2兾9 苷 1,

which we recognize as an equation of an ellipse. In general, the horizontal trace in the plane z 苷 k is x2 ⫹

y2 k2 苷1⫺ 9 4

z苷k

which is an ellipse, provided that k 2 ⬍ 4, that is, ⫺2 ⬍ k ⬍ 2. Similarly, the vertical traces are also ellipses:

z (0, 0, 2)

0 (1, 0, 0)

(0, 3, 0) y

x

FIGURE 4

The ellipsoid ≈+

z@ y@ + =1 4 9

y2 z2 ⫹ 苷 1 ⫺ k2 9 4

x苷k

共if ⫺1 ⬍ k ⬍ 1兲

z2 k2 苷1⫺ 4 9

y苷k

共if ⫺3 ⬍ k ⬍ 3兲

x2 ⫹

Figure 4 shows how drawing some traces indicates the shape of the surface. It’s called an ellipsoid because all of its traces are ellipses. Notice that it is symmetric with respect to each coordinate plane; this is a reflection of the fact that its equation involves only even powers of x, y, and z. M EXAMPLE 4 Use traces to sketch the surface z 苷 4x 2 ⫹ y 2.

SOLUTION If we put x 苷 0, we get z 苷 y 2, so the yz-plane intersects the surface in a

parabola. If we put x 苷 k (a constant), we get z 苷 y 2 ⫹ 4k 2. This means that if we slice the graph with any plane parallel to the yz-plane, we obtain a parabola that opens upward. Similarly, if y 苷 k, the trace is z 苷 4x 2 ⫹ k 2, which is again a parabola that opens upward. If we put z 苷 k, we get the horizontal traces 4x 2 ⫹ y 2 苷 k, which we recognize as a family of ellipses. Knowing the shapes of the traces, we can sketch the graph in Figure 5. Because of the elliptical and parabolic traces, the quadric surface z 苷 4x 2 ⫹ y 2 is called an elliptic paraboloid. z

FIGURE 5 The surface z=4≈+¥ is an elliptic paraboloid. Horizontal traces are ellipses; vertical traces are parabolas.

0 x

y

M

SECTION 12.6 CYLINDERS AND QUADRIC SURFACES

V EXAMPLE 5

||||

807

Sketch the surface z 苷 y 2 ⫺ x 2.

SOLUTION The traces in the vertical planes x 苷 k are the parabolas z 苷 y 2 ⫺ k 2, which

open upward. The traces in y 苷 k are the parabolas z 苷 ⫺x 2 ⫹ k 2, which open downward. The horizontal traces are y 2 ⫺ x 2 苷 k, a family of hyperbolas. We draw the families of traces in Figure 6, and we show how the traces appear when placed in their correct planes in Figure 7. z

z

y

⫾2 0

1

_1

⫾1

_1 0

y

⫾1

FIGURE 6

Vertical traces are parabolas; horizontal traces are hyperbolas. All traces are labeled with the value of k.

x

x

0

⫾2

1 Traces in y=k are z=_≈+k@

Traces in x=k are z=¥-k@

z

Traces in z=k are ¥-≈=k

z

z

1

0 x

_1

x

0

FIGURE 7

x

_1

_1

0

1

Traces moved to their correct planes

gate how traces determine the shape of a surface.

1

Traces in y=k

Traces in x=k

TEC In Module 12.6A you can investi-

y

y

y

Traces in z=k

In Figure 8 we fit together the traces from Figure 7 to form the surface z 苷 y 2 ⫺ x 2, a hyperbolic paraboloid. Notice that the shape of the surface near the origin resembles that of a saddle. This surface will be investigated further in Section 14.7 when we discuss saddle points. z

0 x

FIGURE 8

y

The surface z=¥-≈ is a hyperbolic paraboloid.

M

EXAMPLE 6 Sketch the surface

x2 z2 ⫹ y2 ⫺ 苷 1. 4 4

SOLUTION The trace in any horizontal plane z 苷 k is the ellipse

x2 k2 ⫹ y2 苷 1 ⫹ 4 4

z苷k

808

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CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

z

but the traces in the xz- and yz-planes are the hyperbolas x2 z2 ⫺ 苷1 4 4

y苷0

y2 ⫺

and

z2 苷1 4

x苷0

(0, 1, 0)

(2, 0, 0)

This surface is called a hyperboloid of one sheet and is sketched in Figure 9.

y

x

The idea of using traces to draw a surface is employed in three-dimensional graphing software for computers. In most such software, traces in the vertical planes x 苷 k and y 苷 k are drawn for equally spaced values of k, and parts of the graph are eliminated using hidden line removal. Table 1 shows computer-drawn graphs of the six basic types of quadric surfaces in standard form. All surfaces are symmetric with respect to the z-axis. If a quadric surface is symmetric about a different axis, its equation changes accordingly.

FIGURE 9

TA B L E 1 Graphs of quadric surfaces

Surface

Equation y2 z2 x2 苷1 2 ⫹ 2 ⫹ a b c2

Ellipsoid z

y

x

Equation x2 y2 z2 2 苷 2 ⫹ c a b2

Cone z

All traces are ellipses.

Horizontal traces are ellipses.

If a 苷 b 苷 c, the ellipsoid is a sphere.

Vertical traces in the planes x 苷 k and y 苷 k are hyperbolas if k 苷 0 but are pairs of lines if k 苷 0.

z x2 y2 苷 2 ⫹ 2 c a b

z

y

y2 z2 x2 ⫹ ⫺ 苷1 a2 b2 c2

Hyperboloid of One Sheet z

Horizontal traces are ellipses.

Horizontal traces are ellipses.

Vertical traces are parabolas.

Vertical traces are hyperbolas.

The variable raised to the first power indicates the axis of the paraboloid.

x

Surface

x

Elliptic Paraboloid

x

y

The axis of symmetry corresponds to the variable whose coefficient is negative.

y

z x2 y2 苷 2 ⫺ 2 c a b

Hyperbolic Paraboloid z

Hyperboloid of Two Sheets

⫺

z

Vertical traces are parabolas. The case where c ⬍ 0 is illustrated.

y2 z2 x2 苷1 2 ⫺ 2 ⫹ a b c2

Horizontal traces in z 苷 k are ellipses if k ⬎ c or k ⬍ ⫺c.

Horizontal traces are hyperbolas. y

x

M

Vertical traces are hyperbolas. x

y

The two minus signs indicate two sheets.

SECTION 12.6 CYLINDERS AND QUADRIC SURFACES

TEC In Module 12.6B you can see how changing a, b, and c in Table 1 affects the shape of the quadric surface.

V EXAMPLE 7

||||

809

Identify and sketch the surface 4x 2 ⫺ y 2 ⫹ 2z 2 ⫹ 4 苷 0.

SOLUTION Dividing by ⫺4, we first put the equation in standard form:

⫺x 2 ⫹

y2 z2 ⫺ 苷1 4 2

Comparing this equation with Table 1, we see that it represents a hyperboloid of two sheets, the only difference being that in this case the axis of the hyperboloid is the y-axis. The traces in the xy- and yz-planes are the hyperbolas ⫺x 2 ⫹

y2 苷1 4

z苷0

y2 z2 ⫺ 苷1 4 2

and

x苷0

The surface has no trace in the xz-plane, but traces in the vertical planes y 苷 k for

ⱍ k ⱍ ⬎ 2 are the ellipses

z (0, _2, 0)

x2 ⫹

0

y苷k

which can be written as y

x

z2 k2 苷 ⫺1 2 4

(0, 2, 0)

x2 2

k ⫺1 4

FIGURE 10

⫹

z2

冉 冊 2

k 2 ⫺1 4

苷1

y苷k

These traces are used to make the sketch in Figure 10.

4≈-¥+2z@+4=0

M

EXAMPLE 8 Classify the quadric surface x 2 ⫹ 2z 2 ⫺ 6x ⫺ y ⫹ 10 苷 0.

SOLUTION By completing the square we rewrite the equation as

y ⫺ 1 苷 共x ⫺ 3兲2 ⫹ 2z 2 Comparing this equation with Table 1, we see that it represents an elliptic paraboloid. Here, however, the axis of the paraboloid is parallel to the y-axis, and it has been shifted so that its vertex is the point 共3, 1, 0兲. The traces in the plane y 苷 k 共k ⬎ 1兲 are the ellipses 共x ⫺ 3兲2 ⫹ 2z 2 苷 k ⫺ 1

y苷k

The trace in the xy-plane is the parabola with equation y 苷 1 ⫹ 共x ⫺ 3兲2, z 苷 0. The paraboloid is sketched in Figure 11. z

0 y

FIGURE 11

≈+2z@-6x-y+10=0

x

(3, 1, 0) M

810

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CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

APPLICATIONS OF QUADRIC SURFACES

© Corbis

David Burnett / Photo Researchers, Inc

Examples of quadric surfaces can be found in the world around us. In fact, the world itself is a good example. Although the earth is commonly modeled as a sphere, a more accurate model is an ellipsoid because the earth’s rotation has caused a flattening at the poles. (See Exercise 47.) Circular paraboloids, obtained by rotating a parabola about its axis, are used to collect and reflect light, sound, and radio and television signals. In a radio telescope, for instance, signals from distant stars that strike the bowl are reflected to the receiver at the focus and are therefore amplified. (The idea is explained in Problem 18 on page 268.) The same principle applies to microphones and satellite dishes in the shape of paraboloids. Cooling towers for nuclear reactors are usually designed in the shape of hyperboloids of one sheet for reasons of structural stability. Pairs of hyperboloids are used to transmit rotational motion between skew axes. (The cogs of gears are the generating lines of the hyperboloids. See Exercise 49.)

A satellite dish reflects signals to the focus of a paraboloid.

12.6

Nuclear reactors have cooling towers in the shape of hyperboloids.

Hyperboloids produce gear transmission.

EXERCISES

1. (a) What does the equation y 苷 x 2 represent as a curve in ⺢ 2 ?

(b) What does it represent as a surface in ⺢ 3 ? (c) What does the equation z 苷 y 2 represent?

(b) Sketch the graph of y 苷 e as a surface in ⺢ . (c) Describe and sketch the surface z 苷 e y. 3

3– 8 Describe and sketch the surface. 3. y 2 ⫹ 4z 2 苷 4

4. z 苷 4 ⫺ x 2

6. yz 苷 4

7. z 苷 cos x

8. x 2 ⫺ y 2 苷 1

9. (a) Find and identify the traces of the quadric surface

2. (a) Sketch the graph of y 苷 e x as a curve in ⺢ 2. x

5. x ⫺ y 2 苷 0

x 2 ⫹ y 2 ⫺ z 2 苷 1 and explain why the graph looks like the graph of the hyperboloid of one sheet in Table 1. (b) If we change the equation in part (a) to x 2 ⫺ y 2 ⫹ z 2 苷 1, how is the graph affected? (c) What if we change the equation in part (a) to x 2 ⫹ y 2 ⫹ 2y ⫺ z 2 苷 0?

SECTION 12.6 CYLINDERS AND QUADRIC SURFACES

||||

811

29–36 Reduce the equation to one of the standard forms, classify

10. (a) Find and identify the traces of the quadric surface

⫺x 2 ⫺ y 2 ⫹ z 2 苷 1 and explain why the graph looks like the graph of the hyperboloid of two sheets in Table 1. (b) If the equation in part (a) is changed to x 2 ⫺ y 2 ⫺ z 2 苷 1, what happens to the graph? Sketch the new graph.

the surface, and sketch it. 29. z 2 苷 4x 2 ⫹ 9y 2 ⫹ 36

30. x 2 苷 2y 2 ⫹ 3z 2

31. x 苷 2y 2 ⫹ 3z 2

32. 4x ⫺ y 2 ⫹ 4z 2 苷 0

11–20 Use traces to sketch and identify the surface.

33. 4x 2 ⫹ y 2 ⫹ 4 z 2 ⫺ 4y ⫺ 24z ⫹ 36 苷 0

11. x 苷 y 2 ⫹ 4z 2

12. 9x 2 ⫺ y 2 ⫹ z 2 苷 0

34. 4y 2 ⫹ z 2 ⫺ x ⫺ 16y ⫺ 4z ⫹ 20 苷 0

13. x 2 苷 y 2 ⫹ 4z 2

14. 25x 2 ⫹ 4y 2 ⫹ z 2 苷 100

35. x 2 ⫺ y 2 ⫹ z 2 ⫺ 4x ⫺ 2y ⫺ 2z ⫹ 4 苷 0

15. ⫺x 2 ⫹ 4y 2 ⫺ z 2 苷 4

16. 4x 2 ⫹ 9y 2 ⫹ z 苷 0

36. x 2 ⫺ y 2 ⫹ z 2 ⫺ 2x ⫹ 2y ⫹ 4z ⫹ 2 苷 0

17. 36x 2 ⫹ y 2 ⫹ 36z 2 苷 36

18. 4x 2 ⫺ 16y 2 ⫹ z 2 苷 16

19. y 苷 z 2 ⫺ x 2

20. x 苷 y 2 ⫺ z 2

; 37– 40 Use a computer with three-dimensional graphing software to graph the surface. Experiment with viewpoints and with domains for the variables until you get a good view of the surface.

21–28 Match the equation with its graph (labeled I–VIII). Give reasons for your choices.

37. ⫺4x 2 ⫺ y 2 ⫹ z 2 苷 1

38. x 2 ⫺ y 2 ⫺ z 苷 0

21. x 2 ⫹ 4y 2 ⫹ 9z 2 苷 1

22. 9x 2 ⫹ 4y 2 ⫹ z 2 苷 1

39. ⫺4x 2 ⫺ y 2 ⫹ z 2 苷 0

40. x 2 ⫺ 6x ⫹ 4y 2 ⫺ z 苷 0

23. x 2 ⫺ y 2 ⫹ z 2 苷 1

24. ⫺x 2 ⫹ y 2 ⫺ z 2 苷 1

25. y 苷 2x 2 ⫹ z 2

26. y 2 苷 x 2 ⫹ 2z 2

27. x ⫹ 2z 苷 1

28. y 苷 x ⫺ z

2

2

2

z

I

41. Sketch the region bounded by the surfaces z 苷 sx 2 ⫹ y 2

and x 2 ⫹ y 2 苷 1 for 1 艋 z 艋 2.

2

42. Sketch the region bounded by the paraboloids z 苷 x 2 ⫹ y 2

and z 苷 2 ⫺ x 2 ⫺ y 2.

z

II

43. Find an equation for the surface obtained by rotating the

parabola y 苷 x 2 about the y-axis.

y

x

y

x

44. Find an equation for the surface obtained by rotating the line

x 苷 3y about the x-axis. z

III

z

IV

45. Find an equation for the surface consisting of all points that

are equidistant from the point 共⫺1, 0, 0兲 and the plane x 苷 1. Identify the surface. 46. Find an equation for the surface consisting of all points P for

y

which the distance from P to the x-axis is twice the distance from P to the yz-plane. Identify the surface.

y

x x z

V

y

x

z

VII

y

x

z

VIII

y

y x

47. Traditionally, the earth’s surface has been modeled as a sphere,

z

VI

x

but the World Geodetic System of 1984 (WGS-84) uses an ellipsoid as a more accurate model. It places the center of the earth at the origin and the north pole on the positive z-axis. The distance from the center to the poles is 6356.523 km and the distance to a point on the equator is 6378.137 km. (a) Find an equation of the earth’s surface as used by WGS-84. (b) Curves of equal latitude are traces in the planes z 苷 k. What is the shape of these curves? (c) Meridians (curves of equal longitude) are traces in planes of the form y 苷 mx. What is the shape of these meridians? 48. A cooling tower for a nuclear reactor is to be constructed in

the shape of a hyperboloid of one sheet (see the photo on page 810). The diameter at the base is 280 m and the minimum

812

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CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

diameter, 500 m above the base, is 200 m. Find an equation for the tower. 49. Show that if the point 共a, b, c兲 lies on the hyperbolic paraboloid

z 苷 y 2 ⫺ x 2, then the lines with parametric equations x 苷 a ⫹ t, y 苷 b ⫹ t, z 苷 c ⫹ 2共b ⫺ a兲t and x 苷 a ⫹ t, y 苷 b ⫺ t, z 苷 c ⫺ 2共b ⫹ a兲t both lie entirely on this paraboloid. (This shows that the hyperbolic paraboloid is what is called a ruled surface; that is, it can be generated by the motion of a straight line. In fact, this exercise shows that through each point on the hyperbolic paraboloid there are two

12

generating lines. The only other quadric surfaces that are ruled surfaces are cylinders, cones, and hyperboloids of one sheet.) 50. Show that the curve of intersection of the surfaces

x 2 ⫹ 2y 2 ⫺ z 2 ⫹ 3x 苷 1 and 2x 2 ⫹ 4y 2 ⫺ 2z 2 ⫺ 5y 苷 0 lies in a plane. 2 2 2 ; 51. Graph the surfaces z 苷 x ⫹ y and z 苷 1 ⫺ y on a common

ⱍ ⱍ

ⱍ ⱍ

screen using the domain x 艋 1.2, y 艋 1.2 and observe the curve of intersection of these surfaces. Show that the projection of this curve onto the xy-plane is an ellipse.

REVIEW

CONCEPT CHECK 1. What is the difference between a vector and a scalar?

11. How do you find a vector perpendicular to a plane?

2. How do you add two vectors geometrically? How do you add

12. How do you find the angle between two intersecting planes?

them algebraically? 3. If a is a vector and c is a scalar, how is ca related to a

geometrically? How do you find ca algebraically?

13. Write a vector equation, parametric equations, and symmetric

equations for a line.

4. How do you find the vector from one point to another?

14. Write a vector equation and a scalar equation for a plane.

5. How do you find the dot product a ⴢ b of two vectors if you

15. (a) How do you tell if two vectors are parallel?

know their lengths and the angle between them? What if you know their components? 6. How are dot products useful?

(b) How do you tell if two vectors are perpendicular? (c) How do you tell if two planes are parallel? 16. (a) Describe a method for determining whether three points

7. Write expressions for the scalar and vector projections of b

onto a. Illustrate with diagrams. 8. How do you find the cross product a ⫻ b of two vectors if you

know their lengths and the angle between them? What if you know their components? 9. How are cross products useful? 10. (a) How do you find the area of the parallelogram determined

by a and b? (b) How do you find the volume of the parallelepiped determined by a, b, and c?

P, Q, and R lie on the same line. (b) Describe a method for determining whether four points P, Q, R, and S lie in the same plane. 17. (a) How do you find the distance from a point to a line?

(b) How do you find the distance from a point to a plane? (c) How do you find the distance between two lines? 18. What are the traces of a surface? How do you find them? 19. Write equations in standard form of the six types of quadric

surfaces.

T R U E - FA L S E Q U I Z Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

1. For any vectors u and v in V3 , u ⴢ v 苷 v ⴢ u.

ⱍ

ⱍ ⱍ

u ⴢ 共v ⫻ w兲 苷 共u ⫻ v兲 ⴢ w.

ⱍ

3. For any vectors u and v in V3 , u ⫻ v 苷 v ⫻ u .

k共u ⴢ v兲 苷 共k u兲 ⴢ v. 5. For any vectors u and v in V3 and any scalar k,

k共u ⫻ v兲 苷 共k u兲 ⫻ v.

共u ⫹ v兲 ⫻ w 苷 u ⫻ w ⫹ v ⫻ w. 7. For any vectors u, v, and w in V3,

2. For any vectors u and v in V3 , u ⫻ v 苷 v ⫻ u.

4. For any vectors u and v in V3 and any scalar k,

6. For any vectors u, v, and w in V3,

8. For any vectors u, v, and w in V3 ,

u ⫻ 共v ⫻ w兲 苷 共u ⫻ v兲 ⫻ w. 9. For any vectors u and v in V3 , 共u ⫻ v兲 ⴢ u 苷 0. 10. For any vectors u and v in V3 , 共u ⫹ v兲 ⫻ v 苷 u ⫻ v.

CHAPTER 12 REVIEW

11. The cross product of two unit vectors is a unit vector. 12. A linear equation Ax ⫹ By ⫹ Cz ⫹ D 苷 0 represents a line

in space.

813

15. If u ⴢ v 苷 0 , then u 苷 0 or v 苷 0. 16. If u ⫻ v 苷 0, then u 苷 0 or v 苷 0.

13. The set of points {共x, y, z兲 x 2 ⫹ y 2 苷 1} is a circle.

17. If u ⴢ v 苷 0 , and u ⫻ v 苷 0, then u 苷 0 or v 苷 0.

14. If u 苷 具u1, u2 典 and v 苷 具 v1, v2 典 , then u ⴢ v 苷 具u1v1, u2 v2 典 .

18. If u and v are in V3 , then u ⴢ v 艋 u

ⱍ

||||

ⱍ

ⱍ ⱍ ⱍ ⱍ v ⱍ.

EXERCISES 1. (a) Find an equation of the sphere that passes through the point

共6, ⫺2, 3兲 and has center 共⫺1, 2, 1兲. (b) Find the curve in which this sphere intersects the yz-plane. (c) Find the center and radius of the sphere

2. Copy the vectors in the figure and use them to draw each of the

(c) ⫺ 12 a

(a) 共u ⫻ v兲 ⴢ w (c) v ⴢ 共u ⫻ w兲

(b) u ⴢ 共w ⫻ v兲 (d) 共u ⫻ v兲 ⴢ v

8. Show that if a, b, and c are in V3 , then

共a ⫻ b兲 ⴢ 关共b ⫻ c兲 ⫻ 共c ⫻ a兲兴 苷 关a ⴢ 共b ⫻ c兲兴 2

x 2 ⫹ y 2 ⫹ z 2 ⫺ 8x ⫹ 2y ⫹ 6z ⫹ 1 苷 0 following vectors. (a) a ⫹ b (b) a ⫺ b

7. Suppose that u ⴢ 共v ⫻ w兲 苷 2. Find

(d) 2 a ⫹ b

9. Find the acute angle between two diagonals of a cube. 10. Given the points A共1, 0, 1兲, B共2, 3, 0兲, C共⫺1, 1, 4兲, and

D共0, 3, 2兲, find the volume of the parallelepiped with adjacent edges AB, AC, and AD. 11. (a) Find a vector perpendicular to the plane through the points

a

A共1, 0, 0兲, B共2, 0, ⫺1兲, and C共1, 4, 3兲. (b) Find the area of triangle ABC.

b

12. A constant force F 苷 3 i ⫹ 5 j ⫹ 10 k moves an object along 3. If u and v are the vectors shown in the figure, find u ⴢ v and

ⱍ u ⫻ v ⱍ. Is u ⫻ v directed into the page or out of it?

the line segment from 共1, 0, 2兲 to 共5, 3, 8兲. Find the work done if the distance is measured in meters and the force in newtons.

13. A boat is pulled onto shore using two ropes, as shown in the

diagram. If a force of 255 N is needed, find the magnitude of the force in each rope.

|v|=3 45°

|u|=2

20° 255 N 30°

4. Calculate the given quantity if

a 苷 i ⫹ j ⫺ 2k (a) (c) (e) (g) (i) (k)

b 苷 3i ⫺ 2j ⫹ k

c 苷 j ⫺ 5k

ⱍ ⱍ

2a ⫹ 3b (b) b aⴢb (d) a ⫻ b b⫻c (f) a ⴢ 共b ⫻ c兲 (h) a ⫻ 共b ⫻ c兲 c⫻c ( j) proj a b comp a b The angle between a and b (correct to the nearest degree)

ⱍ

ⱍ

14. Find the magnitude of the torque about P if a 50-N force is

applied as shown. 50 N 30°

5. Find the values of x such that the vectors 具3, 2, x 典 and

具 2x, 4, x 典 are orthogonal.

40 cm

6. Find two unit vectors that are orthogonal to both j ⫹ 2 k

and i ⫺ 2 j ⫹ 3 k.

P

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CHAPTER 12 VECTORS AND THE GEOMETRY OF SPACE

15–17 Find parametric equations for the line. 15. The line through 共4, ⫺1, 2兲 and 共1, 1, 5兲 16. The line through 共1, 0, ⫺1兲 and parallel to the line 1 3

共x ⫺ 4兲 苷 y 苷 z ⫹ 2 1 2

17. The line through 共⫺2, 2, 4兲 and perpendicular to the

plane 2x ⫺ y ⫹ 5z 苷 12

18 –20 Find an equation of the plane. 18. The plane through 共2, 1, 0兲 and parallel to x ⫹ 4y ⫺ 3z 苷 1 19. The plane through 共3, ⫺1, 1兲, 共4, 0, 2兲, and 共6, 3, 1兲 20. The plane through 共1, 2, ⫺2兲 that contains the line

x 苷 2t, y 苷 3 ⫺ t, z 苷 1 ⫹ 3t

21. Find the point in which the line with parametric equations

x 苷 2 ⫺ t, y 苷 1 ⫹ 3t, z 苷 4t intersects the plane 2 x ⫺ y ⫹ z 苷 2. 22. Find the distance from the origin to the line

x 苷 1 ⫹ t, y 苷 2 ⫺ t, z 苷 ⫺1 ⫹ 2t. 23. Determine whether the lines given by the symmetric

equations y⫺2 z⫺3 x⫺1 苷 苷 2 3 4 and

x⫹1 y⫺3 z⫹5 苷 苷 6 ⫺1 2

are parallel, skew, or intersecting. 24. (a) Show that the planes x ⫹ y ⫺ z 苷 1 and

2x ⫺ 3y ⫹ 4z 苷 5 are neither parallel nor perpendicular.

(b) Find, correct to the nearest degree, the angle between these planes. 25. Find an equation of the plane through the line of intersection of

the planes x ⫺ z 苷 1 and y ⫹ 2z 苷 3 and perpendicular to the plane x ⫹ y ⫺ 2z 苷 1. 26. (a) Find an equation of the plane that passes through the points

A共2, 1, 1兲, B共⫺1, ⫺1, 10兲, and C共1, 3, ⫺4兲. (b) Find symmetric equations for the line through B that is perpendicular to the plane in part (a). (c) A second plane passes through 共2, 0, 4兲 and has normal vector 具2, ⫺4, ⫺3典 . Show that the acute angle between the planes is approximately 43⬚. (d) Find parametric equations for the line of intersection of the two planes. 27. Find the distance between the planes 3x ⫹ y ⫺ 4z 苷 2

and 3x ⫹ y ⫺ 4z 苷 24.

28 –36 Identify and sketch the graph of each surface. 28. x 苷 3

29. x 苷 z

30. y 苷 z

31. x 2 苷 y 2 ⫹ 4z 2

2

32. 4x ⫺ y ⫹ 2z 苷 4

33. ⫺4x 2 ⫹ y 2 ⫺ 4z 2 苷 4

34. y 2 ⫹ z 2 苷 1 ⫹ x 2 35. 4x 2 ⫹ 4y 2 ⫺ 8y ⫹ z 2 苷 0 36. x 苷 y 2 ⫹ z 2 ⫺ 2y ⫺ 4z ⫹ 5 37. An ellipsoid is created by rotating the ellipse 4x 2 ⫹ y 2 苷 16

about the x-axis. Find an equation of the ellipsoid. 38. A surface consists of all points P such that the distance from P

to the plane y 苷 1 is twice the distance from P to the point 共0, ⫺1, 0兲. Find an equation for this surface and identify it.

P R O B L E M S P LU S 1. Each edge of a cubical box has length 1 m. The box contains nine spherical balls with the

1m

same radius r. The center of one ball is at the center of the cube and it touches the other eight balls. Each of the other eight balls touches three sides of the box. Thus the balls are tightly packed in the box. (See the figure.) Find r. (If you have trouble with this problem, read about the problem-solving strategy entitled Use Analogy on page 76.) 2. Let B be a solid box with length L , width W, and height H. Let S be the set of all points that

1m

are a distance at most 1 from some point of B. Express the volume of S in terms of L , W, and H.

1m

FIGURE FOR PROBLEM 1

3. Let L be the line of intersection of the planes cx ⫹ y ⫹ z 苷 c and x ⫺ cy ⫹ cz 苷 ⫺1,

where c is a real number. (a) Find symmetric equations for L . (b) As the number c varies, the line L sweeps out a surface S. Find an equation for the curve of intersection of S with the horizontal plane z 苷 t (the trace of S in the plane z 苷 t). (c) Find the volume of the solid bounded by S and the planes z 苷 0 and z 苷 1. 4. A plane is capable of flying at a speed of 180 km兾h in still air. The pilot takes off from an

airfield and heads due north according to the plane’s compass. After 30 minutes of flight time, the pilot notices that, due to the wind, the plane has actually traveled 80 km at an angle 5° east of north. (a) What is the wind velocity? (b) In what direction should the pilot have headed to reach the intended destination? 5. Suppose a block of mass m is placed on an inclined plane, as shown in the figure. The block’s

N

F

W ¨ FIGURE FOR PROBLEM 5

descent down the plane is slowed by friction; if ␪ is not too large, friction will prevent the block from moving at all. The forces acting on the block are the weight W, where W 苷 mt ( t is the acceleration due to gravity); the normal force N (the normal component of the reactionary force of the plane on the block), where N 苷 n; and the force F due to friction, which acts parallel to the inclined plane, opposing the direction of motion. If the block is at rest and ␪ is increased, F must also increase until ultimately F reaches its maximum, beyond which the block begins to slide. At this angle ␪s , it has been observed that F is proportional to n. Thus, when F is maximal, we can say that F 苷 ␮ s n, where ␮ s is called the coefficient of static friction and depends on the materials that are in contact. (a) Observe that N ⫹ F ⫹ W ⫽ 0 and deduce that ␮ s 苷 tan共␪s兲 . (b) Suppose that, for ␪ ⬎ ␪ s , an additional outside force H is applied to the block, horizontally from the left, and let H 苷 h. If h is small, the block may still slide down the plane; if h is large enough, the block will move up the plane. Let h min be the smallest value of h that allows the block to remain motionless (so that F is maximal). By choosing the coordinate axes so that F lies along the x-axis, resolve each force into components parallel and perpendicular to the inclined plane and show that

ⱍ ⱍ

ⱍ ⱍ

ⱍ ⱍ

ⱍ ⱍ

ⱍ ⱍ

ⱍ ⱍ

ⱍ ⱍ

ⱍ ⱍ

ⱍ ⱍ

h min sin ␪ ⫹ mt cos ␪ 苷 n (c) Show that

and

h min cos ␪ ⫹ ␮ s n 苷 mt sin ␪

h min 苷 mt tan共␪ ⫺ ␪s 兲

Does this equation seem reasonable? Does it make sense for ␪ 苷 ␪s ? As ␪ l 90⬚ ? Explain. (d) Let h max be the largest value of h that allows the block to remain motionless. (In which direction is F heading?) Show that h max 苷 m t tan共␪ ⫹ ␪s 兲 Does this equation seem reasonable? Explain.

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