80.1 Vector Properties Concept Overview

VECTOR MECHANICS | CONCEPT OVERVIEW The topic of VECTORS can be referenced on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. VECTOR MECHANICS is a branch of engineering sciences that is concerned with the state of rest, or motion, of bodies subjected to the action of forces. These forces are represented as VECTORS, which characterize a QUANTITY, or MAGNITUDE, as well as a DIRECTION. The fundamentals that are presented in VECTOR MECHANICS will be utilized in our studies of STATICS and DYNAMICS, as well as a number of other subjects within our individual disciplines. A VECTOR is a universal component that can be used to quantify how a FORCE, or ACTION, is applied to a structure, truss, beam or a number of other applications we will encounter. Using the scenario of a golf club striking a golf ball, a few ways that we can apply VECTORS are: • A VELOCITY VECTOR can describe the velocity motion of a golf ball • A DISTANCE VECTOR can help determine how far away and in what direction a golf ball lands • A FORCE VECTOR can describe how hard and in what direction the golf club

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When performing engineering calculations and analysis utilizing VECTOR MECHANICS, we assume that the bodies and particles of interest are RIGID BODIES, and will not deform under load. STATICS is the study of VECTOR MECHANICS that deals with bodies under action of forces that are either at rest or move with a constant velocity. DYNAMICS is the study of motion of bodies under accelerated motion. VECTORS are commonly represented in VECTOR NOTATION, where a SCALAR is used to represent the component of each quantity with respect to a particular axis or direction.

A vector representing 3 components, or dimensions, can be expressed in VECTOR NOTATION as: 𝐴 = 𝑎! 𝑖 + 𝑎! 𝑗 + 𝑎! 𝑘

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Every VECTOR is unique based on a number of CHARACTERISTICS, generally illustrated as:

Let’s highlight some of the more important CHARACTERISTICS of note: • A SCALAR is a mathematical quantity that retains a magnitude only, whereas, a vector is one that possesses both magnitude and direction. • The SENSE of a vector is the SIGN OF THE MAGNITUDE, or the direction in which the vector is acting. The sense is the part of a vector that indicates whether a football thrown is coming towards you or away from you. • The POINT OF APPLICATION is the physical location on the object or in space where the vector is acting. The LINE OF ACTION represents the line space on which the vector is acting. • The HEAD is the vector’s sense and is indicated by the arrowhead.

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• The TAIL of the arrow typically depicts the vector’s point of application. • The SHAFT is the actual line-length of the arrow representing the vector’s magnitude, where a longer vector drawing implies a large action and vice versa. • The LABEL of the vector helps to label or distinguish the vector from other vectors in the analysis. Two vectors are said to be the same if they have the same MAGNITUDE and DIRECTION. However, they can be anywhere in space, and do not need to have the same point of application. A NEGATIVE VECTOR is a vector with the same magnitude, but OPPOSITE DIRECTION.

TYPES OF VECTORS: FIXED – A FIXED (OR BOUND) VECTOR has a UNIQUE POINT OF APPLICATION specified and therefore can’t be moved without modifying the conditions of the problem. An example is the action of a force on a deformable body or the weight vector from the center of gravity of a body: FREE – A FREE VECTOR does not have a CONFINED ACTION or associated with any particular line in space. An example of movement of a body without rotation. These vectors are usually moments and couples that result in a specific action but may be freely moved around the object without changing the original behavior.

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SLIDING – A SLIDING VECTOR has a UNIQUE line in space, or LINE OF ACTION, which must be maintained, but NOT A UNIQUE POINT OF APPLICATION. An example is an external force on a rigid body such as that impulse in dynamics.



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VECTOR ADDITION | CONCEPT OVERVIEW The TOPIC of VECTOR ADDITION can be referenced under the subject of MATHEMATICS on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The ADDITION OF VECTORS involves collecting each of the pieces of the action that are acting in a common direction and then representing them with some indicator of the direction of those pieces. VECTOR ADDITION is defined as calculating the sum between the scalars of each component with respect to component or dimension. The GENERAL FORMULA for VECTOR ADDITION can be referenced under the subject of MATHEMATICS on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Given two vectors, 𝐴 and 𝐵: 𝐴 = 𝑎! 𝑖 + 𝑎! 𝑗 + 𝑎! 𝑘 𝐵 = 𝑏! 𝑖 + 𝑏! 𝑗 + 𝑏! 𝑘 We can write the SUM of the two vectors as: 𝐴 + 𝐵 = 𝑎! + 𝑏! 𝑖 + 𝑎! + 𝑏! 𝑗 + 𝑎! + 𝑏! 𝑘



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PARALLELOGRAM LAW We know that a vector has a MAGNITUDE and DIRECTION, allowing us to represent systems of vectors said to be in equilibrium as triangles, or parallelograms, with defined geometric relationships. This also allows a means to determine the MAGNITUDE and DIRECTION of the ADDITION of two VECTORS. This can be generally illustrated as:

This is how this all comes together. Given two vectors, 𝐴 and 𝐵, we can TRANSLATE, or move, vector “𝐵” until the TAIL matches the HEAD of vector “𝐴”. In this move, we are only TRANSLATING only, the LINE OF ACTION remains the SAME. When this is done, a new line segment can be created from the TAIL of vector “𝐴” to the HEAD of vector “𝐵”. This new line segment represents the SUM of the two vectors, or

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𝐴 + 𝐵. Moving vectors head to tail when adding them is a quick and easy way of analyzing a system of forces or various actions applied to a single body. In fact, if you have a hundred simultaneous actions on an object, connecting the tail of one of the actions to the head of another action, and then repeating this until all the actions are connected with one another, the end result will ultimately be a new line segment from the TAIL of the starting ACTION to the HEAD of the very last ACTION that is put in to place. The COMMUTATIVE LAW, also known as the PARALLELOGRAM LAW, states that the order of addition does not matter. As long as we maintain the direction and orientation of the vectors, we can calculate the sum of the vectors in a variety of ways. The COMMUTATIVE PROPERTY of VECTOR ADDITION is represented by the expression: 𝐴+𝐵 =𝐵+𝐴 The ASSOCIATIVE LAW states that the sum of three vectors does not depend on which pair of vectors is added first. The ASSOCIATIVE PROPERTY of VECTOR ADDITION can be expressed as: 𝐴+ 𝐵+𝐶 = 𝐴+𝐵 +𝐶 Order doesn’t matter for simultaneous events. However, you can only attach a SINGLE TAIL of a vector to any given VECTOR HEAD. You can’t attach the tails of multiple

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vectors to the head of the same vector. These LAWS graphically, we have:

GUIDELINES ON USING THE PARALLELOGRAM LAW: 1. Select an arbitrary starting point. The origin of your coordinate system is often a convenient starting point, but you’re not required to use that point. However, you must choose an origin somewhere. 2. Choose any one of the original vectors and place its tail at the starting location you determined in Step 1, making sure to maintain the original orientation and sense. You can start with either vector, but after you use a vector one time, you can’t use it

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again. You must keep the same magnitude and direction for each vector. 3. Select one of the remaining original vectors and affix the tail of that vector to the head of the vector from Step 2. 4. Keep attaching vectors to each other (by repeating Steps 2 and 3) until you’ve used all of the original vectors. 5. After all vectors have been properly attached and distributed, connect the HEAD of the final vector to the ORIGIN, or the TAIL of the first vector you placed. This represents the CUMMALATIVE EFFECTS of the actions in the context in which they are applied.



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VECTOR ADDITION | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material.

The resultant vector formed by the sum of the following vectors, residing in a typical xy coordinate system, is most close to: 𝐴 = 9,10 𝐵 = 1,4 A. 10𝑖 + 14𝑗 B. 47𝑖 + 𝑗 C. 12𝑖 + 2𝑗 D. 14𝑖 + 10𝑗

SOLUTION: The TOPIC of VECTOR ADDITION can be referenced under the subject of MATHEMATICS on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The ADDITION of VECTORS involves breaking down each element of a given vector and combining those elements as they relate to a common dimension, or direction.



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The result of combining the elements of a pair of vectors provides us with a new unique vector. VECTOR ADDITION is defined as calculating the sum between the scalars of each component with respect to each component or dimension. The GENERAL FORMULA for VECTOR ADDITION can be referenced under the topic of MATHEMATICS on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Given two vectors, 𝐴 and 𝐵: 𝐴 = 𝑎! 𝑖 + 𝑎! 𝑗 + 𝑎! 𝑘 𝐵 = 𝑏! 𝑖 + 𝑏! 𝑗 + 𝑏! 𝑘 We can write the SUM of the two vectors as: 𝐴 + 𝐵 = 𝑎! + 𝑏! 𝑖 + 𝑎! + 𝑏! 𝑗 + 𝑎! + 𝑏! 𝑘 In this problem, we are given: 𝐴 = 9,10 𝐵 = 1,4 These VECTORS reside in your typical x-y coordinate system, as defined in the problem statement.

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Writing them both using VECTOR NOTATION, we have: 𝐴 = 9𝑖 + 10𝑗 𝐵 = 1𝑖 + 4𝑗 Recall the COMMUTATIVE LAW states that the order of addition does not matter, we could plug in the elements in whatever order, such that: 𝐴+𝐵 =𝐵+𝐴 We can write the SUM of the two vectors as: 𝐴 + 𝐵 = 𝑎! + 𝑏! 𝑖 + 𝑎! + 𝑏! 𝑗 As you can see, we have our VECTORS broken down in to their individual ELEMENTS, the x and y terms (i and j) and we are just combining them using the ADDITION property. Plugging in the appropriate values we have: 𝐶 = 𝐴 + 𝐵 = 9 + 1 𝑖 + 10 + 4 𝑗 Which gives us the ELEMENTS of our new vector in VECTOR NOTATION as: 𝐶 = 𝐴 + 𝐵 = 10𝑖 + 14𝑗 The correct answer choice is A. 𝟏𝟎𝒊 + 𝟏𝟒𝒋

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VECTOR SUBTRACTION | CONCEPT OVERVIEW The TOPIC of VECTOR SUBTRACTION can be referenced under the subject of MATHEMATICS on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. VECTOR SUBSTRACTION is defined as calculating the DIFFERENCE between the scalars of each component with respect to the dimension in which they reside. The SUBTRACTION OF VECTORS is the reverse operation of the additional of vectors. The only difference is that you actually convert the vector being subtracted to negative vector and then add the vectors. To create a negative vector, you just need to reverse the signs of each of the scalar coefficient, such that:

When SUBTRACTING VECTORS, we must be careful when using geometric methods to help us get to the result. The biggest element being defining the positive and negative directions, such that the difference represents the magnitude of the vector in each direction.

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Graphically, we can illustrate VECTOR SUBTRACTION generically as:

This is how this all comes together. Given two vectors, 𝐴 and 𝐵, we can TRANSLATE, or move, vector “𝐵” until the TAIL matches the TAIL of vector “𝐴”. In this move, we are only TRANSLATING only, the LINE OF ACTION remains the SAME. When this is done, a new line segment can be created from the HEAD of vector “𝐴” to the HEAD of vector “𝐵”. This new line segment represents the SUBTRACTION of the two vectors, or 𝐴 − 𝐵. Defining the equation for the SUBTRACTION of VECTORS, we can write it as such: 𝐵 − 𝐴 = 𝐵 + (−𝐴)



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The GENERAL FORMULA for VECTOR SUBTRACTION can be referenced under the subject of MATHEMATICS on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Given two vectors, 𝐴 and 𝐵, the difference, or the SUBTRACTION, of the two can be written as: 𝐴 − 𝐵 = 𝑎! − 𝑏! 𝑖 + 𝑎! − 𝑏! 𝑗 + 𝑎! − 𝑏! 𝑘



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VECTOR SUBTRACTION | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material.

The difference between the vectors 𝐾 and 𝑉 expressed in vector notation as vector 𝑋, is most close to: 𝐾 = 4𝑖 + 5𝑗

𝑉 = 12𝑖 + 2𝑗

A. 𝑋 = 9𝑖 − 14𝑗 B. 𝑋 = −3𝑖 + 8𝑗 C. 𝑋 = −8𝑖 + 3𝑗 D. 𝑋 = 8𝑖 + 3𝑗

SOLUTION: The TOPIC of VECTOR SUBTRACTION can be referenced under the subject of MATHEMATICS on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. VECTOR SUBSTRACTION is defined as calculating the DIFFERENCE between the scalars of each component with respect to the dimension in which they reside.



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In this problem, we are given the two VECTORS: 𝐾 = 4𝑖 + 5𝑗 𝑉 = 12𝑖 + 2𝑗 And we are asked to determine the DIFFERENCE between the two, therefore, we are dealing with VECTOR SUBTRACTION. The SUBTRACTION of VECTORS is the reverse operation of the additional of vectors. The only difference is that you actually convert the vector being subtracted to negative vector and then add the vectors. The OPERATION that we are asked to carry out is: 𝑋 =𝐾−𝑉 Which is equivalent to the OPERATION: 𝑋 = 𝐾 + (−𝑉) This illustrates how the SUBTRACTION of VECTORS is just the reverse operation of the additional of vectors. So let’s create the NEGATIVE VECTOR for VECTOR V. If:



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𝑉 = 12𝑖 + 2𝑗 Then: −𝑉 = −12𝑖 − 2𝑗 Creating this NEGATIVE VECTOR isn’t really necessary, but for illustration and edification purposes we did. The GENERAL FORMULA for VECTOR SUBTRACTION can be referenced under the subject of MATHEMATICS on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Given two vectors, 𝐾 and 𝑉, the difference, or the SUBTRACTION, of the two can be written as: 𝐾 − 𝑉 = 𝑘! − 𝑣! 𝑖 + 𝑘! − 𝑣! 𝑗 + 𝑘! − 𝑣! 𝑘 Plugging in our ELEMENTS we get: 𝐾 − 𝑉 = 4 − 12 𝑖 + 5 − 2 𝑗 Which gives us the ELEMENTS of our new vector in VECTOR NOTATION as: 𝑋 = 𝐾 − 𝑉 = −8𝑖 + 3𝑗 The correct answer choice is C. 𝑿 = −𝟖𝒊 + 𝟑𝒋

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SCALAR MULTIPLICATION | CONCEPT OVERVIEW The GENERAL FORMULA for SCALAR MULTIPLICATION is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We must memorize this formula and understand its application independent of the NCEES Supplied Reference Handbook. A SCALAR is a mathematical quantity that retains a MAGNITUDE ONLY, whereas, a vector is one that possesses both magnitude and direction. SCALAR MULTIPLICATION is the multiplication of a vector by a scalar. When working these types of problems, think of it as SCALING UP or SCALING DOWN the MAGNITUDE of a given vector. Carrying out SCALAR MULTIPLICATION with a POSITIVE value other than 1 changes the magnitude of the vector by that SCALE, but not its direction. Carrying our SCALAR MULTIPLICATION by a NEGATIVE value equal to or less than 1 changes the MAGNITUDE of the vector by that SCALE and REVERSES it direction. The DISTRIBUTIVE PROPERTY of SCALAR MULTIPLICATION is represented by the expression: 𝑎 𝐴 + 𝐵 = 𝑎𝐴 + 𝑎𝐵 Where 𝑎 is a SCALAR VALUE.

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SCALAR MULTIPLICATION | CONCEPT OVERVIEW The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material.

Multiplying vector M by the scalar 3 results in a vector written as: 𝑀 = 7𝑖 + 3𝑗 A. 9𝑖 + 21𝑗 B. 21𝑖 + 9𝑗 C. 21𝑖 + 3𝑗 D. 7𝑖 + 9𝑗

SOLUTION: The GENERAL FORMULA for SCALAR MULTIPLICATION is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We must memorize this formula and understand its application independent of the NCEES Supplied Reference Handbook. A SCALAR is a mathematical quantity that retains a magnitude only, whereas, a vector is one that possesses both magnitude and direction.



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In this problem, we are given a VECTOR: 𝑀 = 7𝑖 + 3𝑗 And a SCALAR with a MAGNITUDE of 3. Carrying out SCALAR MULTIPLICATION with a POSITIVE value other than 1 changes the MAGNITUDE of the vector by that SCALE, but not its direction. Carrying our SCALAR MULTIPLICATION by a NEGATIVE value equal to or less than 1 changes the MAGNITUDE of the vector by that SCALE and REVERSES it direction. In this problem, we will be SCALING UP the MAGNITUDE, but maintaining the DIRECTION. We can deploy the DISTRIBUTIVE PROPERTY of SCALAR MULTIPLICATION to define our result, which is expressed as: 𝑎 𝐴 + 𝐵 = 𝑎𝐴 + 𝑎𝐵 Or if we are dealing with only a single vector: 𝑎 𝐴 = 𝑎 𝑎! 𝑖 + 𝑎! 𝑗 + 𝑎! 𝑘 Plugging in our data, we get: 𝑎 𝑀 =3 7 𝑖+ 3 𝑗

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Note that we changed the NOMENCLATURE on the left side of the expression to match the variable of the VECTOR that we are given, which is VECTOR M. This has zero IMPACT on the overall calculation of the problem, that variable can literally be anything, even a poop emoji if you wanted to throw a little humor in to it…do it. And DISTRIBUTING our SCALAR THROUGH: 𝑎 𝑀 =3 7 𝑖+3 3 𝑗 Resulting in the SCALED UP version of our ORIGNAL VECTOR: 𝑀! = 21𝑖 + 9𝑗 The biggest error we see in these types of problems is when a student distributes the scalar to only a single element of the VECTOR. The correct answer choice is B. 𝟐𝟏𝒊 + 𝟗𝒋



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