Calculus: introduction to lines and planes by: javier
Calculus: introduction to lines and planes
by: javier
RR
RR the cross product
the cross product
□ define the cross: [2, 3, 0] × [1, 0, −1]
the cross product
□ define the cross: [2, 3, 0] × [1, 0, −1] □ perpendicular to each
the cross product
□ define the cross: [2, 3, 0] × [1, 0, −1] □ perpendicular to each □ ||u × v|| is the area
the cross product
□ define the cross: [2, 3, 0] × [1, 0, −1] □ perpendicular to each □ ||u × v|| is the area □ ||u × v|| is ||u||||v|| sin(θ)
the cross product
□ define the cross: [2, 3, 0] × [1, 0, −1] □ perpendicular to each □ ||u × v|| is the area □ ||u × v|| is ||u||||v|| sin(θ) □ right-hand rule
the cross product
□ define the cross: [2, 3, 0] × [1, 0, −1] □ perpendicular to each □ ||u × v|| is the area □ ||u × v|| is ||u||||v|| sin(θ) □ right-hand rule □ test for parallel
the cross product
□ define the cross: [2, 3, 0] × [1, 0, −1] □ perpendicular to each □ ||u × v|| is the area □ ||u × v|| is ||u||||v|| sin(θ) □ right-hand rule □ test for parallel □ volumes of boxes
the cross product
□ define the cross: [2, 3, 0] × [1, 0, −1] □ perpendicular to each □ ||u × v|| is the area □ ||u × v|| is ||u||||v|| sin(θ) □ right-hand rule □ test for parallel □ volumes of boxes
Lines
Lines Old Way of Graphing
Old Way of Graphing
example: plot y − 3 =
3 (x − 1) 2
Old Way of Graphing
example: plot y − 3 =
3 (x − 1) 2
6 5 4 3 2 1
−3
−2
−1 −1 −2 −3
1
2
3
4
5
6
Lines
Lines New Way of Graphing: L(t) = ⃗Pstart + t · ⃗Ndirection
New Way of Graphing: L(t) = ⃗Pstart + t · ⃗Ndirection
New Way of Graphing: L(t) = ⃗Pstart + t · ⃗Ndirection
3 (x − 1) becomes L(t) = ⟨1, 3⟩ + t⟨2, 3⟩ 2
y−3= 6 5 4 3 2 1
−3
−2
−1 −1 −2 −3
1
2
3
4
5
6
New Way of Graphing: L(t) = ⃗Pstart + t · ⃗Ndirection
L(t) = ⟨−2, 3⟩ + t⟨4, −1⟩ 6
5 4 3 2 1
−3
−2
−1 −1 −2 −3
1
2
3
4
5
6
New Way of Graphing: L(t) = ⃗Pstart + t · ⃗Ndirection
L(t) = ⟨2, 3, 1⟩ + t⟨4, −1, 2⟩ 6
5 4 3 2 1
−3 −2 −1 −1 −2 −3
1
2
3
4
5
6
New Way of Graphing: L(t) = ⃗Pstart + t · ⃗Ndirection
L(t) = ⟨−2, 1, 4⟩ + t⟨3, 1, 1⟩ 6
5 4 3 2 1
−3 −2 −1 −1 −2 −3
1
2
3
4
5
6
Planes
Planes basic graphing of planes
basic graphing of planes
graph x6 + y + z = 5 5 4 3 2 1
−3 −2 −1 −1 −2 −3
1
2
3
4
5
6
basic graphing of planes
graph x6 + y − z = −3 5 4 3 2 1
−3 −2 −1 −1 −2 −3
1
2
3
4
5
6
basic graphing of planes
graph 2x + 4y − 3z = 12 6 5 4 3 2 1
−3 −2 −1 −1 −2 −3
1
2
3
4
5
6
Planes
Planes equations of planes
equations of planes
□ every plane has a normal
equations of planes
□ every plane has a normal □ any two points ⃗v = (x, y, z) and ⃗p = (a, b, c) on the plane make a new vector,⃗v − ⃗p, parallel to the plane.
equations of planes
□ every plane has a normal □ any two points ⃗v = (x, y, z) and ⃗p = (a, b, c) on the plane make a new vector,⃗v − ⃗p, parallel to the plane. □ every vector parallel to the plane is perpendicular to the normal of the plane.
equations of planes
□ every plane has a normal □ any two points ⃗v = (x, y, z) and ⃗p = (a, b, c) on the plane make a new vector,⃗v − ⃗p, parallel to the plane. □ every vector parallel to the plane is perpendicular to the normal of the plane. □ therefore (⃗v − ⃗p) · ⃗n = 0
equations of planes
□ every plane has a normal □ any two points ⃗v = (x, y, z) and ⃗p = (a, b, c) on the plane make a new vector,⃗v − ⃗p, parallel to the plane. □ every vector parallel to the plane is perpendicular to the normal of the plane. □ therefore (⃗v − ⃗p) · ⃗n = 0 n1 (x − a) + n2 (y − b) + n3 (z − c) = 0
give the equation of such plane
Planes
Planes write equations of planes
write equations of planes □ write the equation of the plane with normal [1, 3, −1] and containing point (9, 2, 7)
write equations of planes
□ write the equation of the plane with normal [−2, −3, −1] and containing point (1, 0, 3)
write equations of planes
□ write the equation of the plane with normal [2, 0, −1] and containing point (1, 0, −2)
Planes
Planes some typical questions
some typical questions
□ find the equation of the plane through (3,2,1) parallel to 3x + 2y + z = 10
some typical questions □ find the shortest distance from point (5, 6, 4) to the plane 3x + 2y + z = 10
some typical questions
□ find the equation of the line where planes 3x + 2y + z = 10 and x + 2y + 5z = 10 intersect
by: javier
RR
RR the cross product
the cross product
□ define the cross: [2, 3, 0] × [1, 0, −1]
the cross product
□ define the cross: [2, 3, 0] × [1, 0, −1] □ perpendicular to each
the cross product
□ define the cross: [2, 3, 0] × [1, 0, −1] □ perpendicular to each □ ||u × v|| is the area
the cross product
□ define the cross: [2, 3, 0] × [1, 0, −1] □ perpendicular to each □ ||u × v|| is the area □ ||u × v|| is ||u||||v|| sin(θ)
the cross product
□ define the cross: [2, 3, 0] × [1, 0, −1] □ perpendicular to each □ ||u × v|| is the area □ ||u × v|| is ||u||||v|| sin(θ) □ right-hand rule
the cross product
□ define the cross: [2, 3, 0] × [1, 0, −1] □ perpendicular to each □ ||u × v|| is the area □ ||u × v|| is ||u||||v|| sin(θ) □ right-hand rule □ test for parallel
the cross product
□ define the cross: [2, 3, 0] × [1, 0, −1] □ perpendicular to each □ ||u × v|| is the area □ ||u × v|| is ||u||||v|| sin(θ) □ right-hand rule □ test for parallel □ volumes of boxes
the cross product
□ define the cross: [2, 3, 0] × [1, 0, −1] □ perpendicular to each □ ||u × v|| is the area □ ||u × v|| is ||u||||v|| sin(θ) □ right-hand rule □ test for parallel □ volumes of boxes
Lines
Lines Old Way of Graphing
Old Way of Graphing
example: plot y − 3 =
3 (x − 1) 2
Old Way of Graphing
example: plot y − 3 =
3 (x − 1) 2
6 5 4 3 2 1
−3
−2
−1 −1 −2 −3
1
2
3
4
5
6
Lines
Lines New Way of Graphing: L(t) = ⃗Pstart + t · ⃗Ndirection
New Way of Graphing: L(t) = ⃗Pstart + t · ⃗Ndirection
New Way of Graphing: L(t) = ⃗Pstart + t · ⃗Ndirection
3 (x − 1) becomes L(t) = ⟨1, 3⟩ + t⟨2, 3⟩ 2
y−3= 6 5 4 3 2 1
−3
−2
−1 −1 −2 −3
1
2
3
4
5
6
New Way of Graphing: L(t) = ⃗Pstart + t · ⃗Ndirection
L(t) = ⟨−2, 3⟩ + t⟨4, −1⟩ 6
5 4 3 2 1
−3
−2
−1 −1 −2 −3
1
2
3
4
5
6
New Way of Graphing: L(t) = ⃗Pstart + t · ⃗Ndirection
L(t) = ⟨2, 3, 1⟩ + t⟨4, −1, 2⟩ 6
5 4 3 2 1
−3 −2 −1 −1 −2 −3
1
2
3
4
5
6
New Way of Graphing: L(t) = ⃗Pstart + t · ⃗Ndirection
L(t) = ⟨−2, 1, 4⟩ + t⟨3, 1, 1⟩ 6
5 4 3 2 1
−3 −2 −1 −1 −2 −3
1
2
3
4
5
6
Planes
Planes basic graphing of planes
basic graphing of planes
graph x6 + y + z = 5 5 4 3 2 1
−3 −2 −1 −1 −2 −3
1
2
3
4
5
6
basic graphing of planes
graph x6 + y − z = −3 5 4 3 2 1
−3 −2 −1 −1 −2 −3
1
2
3
4
5
6
basic graphing of planes
graph 2x + 4y − 3z = 12 6 5 4 3 2 1
−3 −2 −1 −1 −2 −3
1
2
3
4
5
6
Planes
Planes equations of planes
equations of planes
□ every plane has a normal
equations of planes
□ every plane has a normal □ any two points ⃗v = (x, y, z) and ⃗p = (a, b, c) on the plane make a new vector,⃗v − ⃗p, parallel to the plane.
equations of planes
□ every plane has a normal □ any two points ⃗v = (x, y, z) and ⃗p = (a, b, c) on the plane make a new vector,⃗v − ⃗p, parallel to the plane. □ every vector parallel to the plane is perpendicular to the normal of the plane.
equations of planes
□ every plane has a normal □ any two points ⃗v = (x, y, z) and ⃗p = (a, b, c) on the plane make a new vector,⃗v − ⃗p, parallel to the plane. □ every vector parallel to the plane is perpendicular to the normal of the plane. □ therefore (⃗v − ⃗p) · ⃗n = 0
equations of planes
□ every plane has a normal □ any two points ⃗v = (x, y, z) and ⃗p = (a, b, c) on the plane make a new vector,⃗v − ⃗p, parallel to the plane. □ every vector parallel to the plane is perpendicular to the normal of the plane. □ therefore (⃗v − ⃗p) · ⃗n = 0 n1 (x − a) + n2 (y − b) + n3 (z − c) = 0
give the equation of such plane
Planes
Planes write equations of planes
write equations of planes □ write the equation of the plane with normal [1, 3, −1] and containing point (9, 2, 7)
write equations of planes
□ write the equation of the plane with normal [−2, −3, −1] and containing point (1, 0, 3)
write equations of planes
□ write the equation of the plane with normal [2, 0, −1] and containing point (1, 0, −2)
Planes
Planes some typical questions
some typical questions
□ find the equation of the plane through (3,2,1) parallel to 3x + 2y + z = 10
some typical questions □ find the shortest distance from point (5, 6, 4) to the plane 3x + 2y + z = 10
some typical questions
□ find the equation of the line where planes 3x + 2y + z = 10 and x + 2y + 5z = 10 intersect
