40. 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. A VECTOR is a directed magnitude or quantity that has both magnitude and direction. Vectors are commonly used to represents quantities with directions such as forces, velocities, and electromagnetic forces. 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. We will utilize vector mechanics for our studies of STATICS and DYNAMICS. A VECTOR is a quantity that helps describe the way that a force or action is applied to a structure, truss, beam or one of the numerous other applications we will study. For example, a velocity vector can describe the velocity motion of a golf ball after it has been hit by a nine-iron, and a distance vector can help depict how far away and in what direction it landed. A force vector can describe how hard and in what direction the golf club strikes the golf ball. 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.
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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 vector notation as: π΄ = π$ π + π' π + π) π 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.
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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 vectorβs sense and is indicated by the arrowhead. The TAIL of the arrow typically depicts the vectorβs point of application. The SHAFT is the actual line-length of the arrow representing the vectorβs 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.
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Two vectors are the same if they have the same magnitude and direction. When given two vectors, they are said to be equal if they maintain 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 β A fixed (or bound) vector has a unique point of application is specified and therefore canβt 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 β 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. Sliding β 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: The TOPIC of 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. 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 places. VECTOR ADDITION is defined as calculating the sum between the scalars of each component with respect to component or dimension. The 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, π΄ πππ π΅, can write the sum of the two vectors as: π΄ + π΅ = π$ + π$ π + π' + π' π + π) + π) π As we know that a vector has a magnitude and direction, we can represent systems of vectors as triangles or parallelograms, which have defined geometric relationships we can use. Given two vectors, π΄ πππ π΅, we can calculate the sum of the two vectors by stating π΄ + π΅. We then translate, or move vector βπ΅β until the tail matches the head of vector βπ΄β.
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Then, a new line segment can be created from the tail of vector βπ΄β to the head of vector βπ΅β. This new line segment is the sum of the two vectors or π΄ + π΅.
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 body. In fact, if you have a hundred simultaneous actions on an object, connecting the tail of one action to the head of another action for every action on the object helps you determine the combined response. The final combined response will be from the tail of the very first action you listed to the head of the very last action you listed. 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: π΄+π΅ =π΅+π΄
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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: π΄+ π΅+πΆ = π΄+π΅ +πΆ Order doesnβt matter for simultaneous events. However, you can only attach a single tail of a vector to any given vector arrowhead. You canβt attach the tails of multiple vectors to the head of the same vector.
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Guidelines on Using the Parallelogram Law: 1. Select an arbitrary starting point. β¨ The origin of your coordinate system is often a convenient starting point, but youβre 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 canβt use it again. You must keep the same magnitude and direction for each vector. β¨ 3.Select one of the remaining original vectors and affix the tail of that vector to the head of the vector from Step 2. β¨ 4. Keep attaching vectors to each other (by repeating Steps 2 and 3) until youβve used all of the original vectors. β¨
CONCEPT EXAMPLE: Find the sum, or resultant, of the following vectors: π΄ = 9,10 πππ π΅ = 1,4 A. (10, 14) B. (41, 7) C. (12, 2) D. (2, 2
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SOLUTION: The 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, π΄ πππ π΅, can write the sum of the two vectors as: π΄ + π΅ = π$ + π$ π + π' + π' π + π) + π) π Recall the commutative law that states that the order of addition does not matter. π΄+π΅ =π΅+π΄ As such, it does matter how we add π΄ + π΅ as long as we add like terms. The βπ₯β terms are added together, and then the βπ¦β terms are added together in order to find the resultant vector of βπΆβ. So, letβs add the like terms. πΆ = π΄ + π΅ = 9,10 + 1,4 = 9 + 1, 10 + 4 = (10, 14) Therefore, the sum of the two vectors, π΄ + π΅ is πΆ = (10,14)
Therefore, the correct answer choice is A. (ππ, ππ).
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VECTOR SUBTRACTION: The TOPIC of VECTOR SUBTRACTION can be referenced under the topic 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 component or dimension.
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. When we are subtracting vectors, we must be careful when using geometric methods to help us calculate the difference. We must make sure that define positive and negative directions, such that the difference represents the magnitude of the vector in each direction.
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We can define the equation for the subtraction of vectors as the following: π΅ β π΄ = π΅ + (βπ΄) The FORMULA FOR VECTOR SUBTRACTION 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, π΄ πππ π΅, can write the difference of the two vectors as: π΄ β π΅ = π$ β π$ π + π' β π' π + π) β π) π
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CONCEPT EXAMPLE: What is the difference between the following vectors πΎ β π expressed in vector notation as vector π? πΎ = 4π + 5π π = 12π + 2π A. π = 9π β 14π B. π = β4π + 8π C. π = 8π β 3π D. π = 8π + 3π
SOLUTION: The FORMULA FOR VECTOR SUBTRACTION 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, π΄ πππ π΅, can write the difference of the two vectors as: π΄ β π΅ = π$ β π$ π + π' β π' π + π) β π) π In this problem, we are given: π = π β πΎ = π‘βπ π’πππππ€π π£πππ’π π€π πππ π πππ£πππ πΎ = 4π + 5π
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π = 12π + 2π Plugging these values in to the subtraction of vectors representation, we get: π = π β πΎ = π$ β π$ π + π' β π' π π = π β πΎ = 12 β 4 π + 2 β 5 π π = π β πΎ = 8π + β3 π π = π β πΎ = 8π β 3π
Therefore, the correct answer choice is C. πΏ = ππ β ππ SCALAR MULTIPLICATION: The 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. A scalar represents values such as volume and mass energy, while a vector represents values such a forces and displacement.
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A scalar has a MAGNITUDE, but not a direction. Examples of scalars are time, volume, density, speed, energy, and mass. The magnitude is the numerical value of a given vector. Constructing a vector requires actually knowing a scalar quantity, which is the magnitude SCALAR MULTIPLICATION is the multiplication of a vector by a scalar. When working these problems, thinking of the magnitude of the scalar, as SCALING UP or SCALING DOWN the vector. Scalar multiplication by a positive number other than 1 changes the magnitude of the vector, but not its direction. Scalar multiplication by β1 reverses it direction but doesnβt change its magnitude. Scalar multiplication by any negative number other than 1 both reverses the direction of the vector and changes it magnitude. The Distributive Property of Vector Addition is represented by the expression: π π΄ + π΅ = ππ΄ + ππ΅ where π is a scalar value
CONCEPT EXAMPLE: Perform the following vector operation of multiplying vector M by the scalar 3: π = 7π + 3π A. 9π + 21π B. 21π + 9π C. 21π + 3π D. 7π + 9π
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SOLUTION: The 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. The goal is to scale-up vector π(πβ) by the scalar 3. We can determine this using the concept of Multiplication of Vectors by a Scalar. When you multiply a vector by a scalar it is called "scaling" a vector, because you change how big or small the vector is. The numbers are called "scalars", because they "scale" the vector up or down. The rectangular form vector π΄ = π₯π + π¦π multiplied by the scalar a is ππ΄ = ππ₯π + ππ¦π In this problem, we are given: ππππππ β π’π π£πππ‘ππ π = πβ = π‘βπ π’πππππ€π π£πππ’π π€π πππ π πππ£πππ π = 7π + 3π π = 3 Plugging these values in to the multiplication of vectors by a scalar representation, we get: aM = M ] = axi + ayj
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3π = π] = 3 7 π + 3 3 π π] = 21π + 9π It is important to make sure that you distribute the scalar value to both of the π πππ π components. In this case, the scalar is only multiplied to the π component.
Therefore, the correct answer choice is B. πππ + ππ
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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 vector notation as: π΄ = π$ π + π' π + π) π 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.
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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 vectorβs sense and is indicated by the arrowhead. The TAIL of the arrow typically depicts the vectorβs point of application. The SHAFT is the actual line-length of the arrow representing the vectorβs 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.
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Two vectors are the same if they have the same magnitude and direction. When given two vectors, they are said to be equal if they maintain 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 β A fixed (or bound) vector has a unique point of application is specified and therefore canβt 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 β 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. Sliding β 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: The TOPIC of 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. 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 places. VECTOR ADDITION is defined as calculating the sum between the scalars of each component with respect to component or dimension. The 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, π΄ πππ π΅, can write the sum of the two vectors as: π΄ + π΅ = π$ + π$ π + π' + π' π + π) + π) π As we know that a vector has a magnitude and direction, we can represent systems of vectors as triangles or parallelograms, which have defined geometric relationships we can use. Given two vectors, π΄ πππ π΅, we can calculate the sum of the two vectors by stating π΄ + π΅. We then translate, or move vector βπ΅β until the tail matches the head of vector βπ΄β.
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Then, a new line segment can be created from the tail of vector βπ΄β to the head of vector βπ΅β. This new line segment is the sum of the two vectors or π΄ + π΅.
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 body. In fact, if you have a hundred simultaneous actions on an object, connecting the tail of one action to the head of another action for every action on the object helps you determine the combined response. The final combined response will be from the tail of the very first action you listed to the head of the very last action you listed. 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: π΄+π΅ =π΅+π΄
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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: π΄+ π΅+πΆ = π΄+π΅ +πΆ Order doesnβt matter for simultaneous events. However, you can only attach a single tail of a vector to any given vector arrowhead. You canβt attach the tails of multiple vectors to the head of the same vector.
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Guidelines on Using the Parallelogram Law: 1. Select an arbitrary starting point. β¨ The origin of your coordinate system is often a convenient starting point, but youβre 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 canβt use it again. You must keep the same magnitude and direction for each vector. β¨ 3.Select one of the remaining original vectors and affix the tail of that vector to the head of the vector from Step 2. β¨ 4. Keep attaching vectors to each other (by repeating Steps 2 and 3) until youβve used all of the original vectors. β¨
CONCEPT EXAMPLE: Find the sum, or resultant, of the following vectors: π΄ = 9,10 πππ π΅ = 1,4 A. (10, 14) B. (41, 7) C. (12, 2) D. (2, 2
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SOLUTION: The 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, π΄ πππ π΅, can write the sum of the two vectors as: π΄ + π΅ = π$ + π$ π + π' + π' π + π) + π) π Recall the commutative law that states that the order of addition does not matter. π΄+π΅ =π΅+π΄ As such, it does matter how we add π΄ + π΅ as long as we add like terms. The βπ₯β terms are added together, and then the βπ¦β terms are added together in order to find the resultant vector of βπΆβ. So, letβs add the like terms. πΆ = π΄ + π΅ = 9,10 + 1,4 = 9 + 1, 10 + 4 = (10, 14) Therefore, the sum of the two vectors, π΄ + π΅ is πΆ = (10,14)
Therefore, the correct answer choice is A. (ππ, ππ).
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VECTOR SUBTRACTION: The TOPIC of VECTOR SUBTRACTION can be referenced under the topic 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 component or dimension.
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. When we are subtracting vectors, we must be careful when using geometric methods to help us calculate the difference. We must make sure that define positive and negative directions, such that the difference represents the magnitude of the vector in each direction.
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We can define the equation for the subtraction of vectors as the following: π΅ β π΄ = π΅ + (βπ΄) The FORMULA FOR VECTOR SUBTRACTION 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, π΄ πππ π΅, can write the difference of the two vectors as: π΄ β π΅ = π$ β π$ π + π' β π' π + π) β π) π
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CONCEPT EXAMPLE: What is the difference between the following vectors πΎ β π expressed in vector notation as vector π? πΎ = 4π + 5π π = 12π + 2π A. π = 9π β 14π B. π = β4π + 8π C. π = 8π β 3π D. π = 8π + 3π
SOLUTION: The FORMULA FOR VECTOR SUBTRACTION 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, π΄ πππ π΅, can write the difference of the two vectors as: π΄ β π΅ = π$ β π$ π + π' β π' π + π) β π) π In this problem, we are given: π = π β πΎ = π‘βπ π’πππππ€π π£πππ’π π€π πππ π πππ£πππ πΎ = 4π + 5π
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π = 12π + 2π Plugging these values in to the subtraction of vectors representation, we get: π = π β πΎ = π$ β π$ π + π' β π' π π = π β πΎ = 12 β 4 π + 2 β 5 π π = π β πΎ = 8π + β3 π π = π β πΎ = 8π β 3π
Therefore, the correct answer choice is C. πΏ = ππ β ππ SCALAR MULTIPLICATION: The 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. A scalar represents values such as volume and mass energy, while a vector represents values such a forces and displacement.
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A scalar has a MAGNITUDE, but not a direction. Examples of scalars are time, volume, density, speed, energy, and mass. The magnitude is the numerical value of a given vector. Constructing a vector requires actually knowing a scalar quantity, which is the magnitude SCALAR MULTIPLICATION is the multiplication of a vector by a scalar. When working these problems, thinking of the magnitude of the scalar, as SCALING UP or SCALING DOWN the vector. Scalar multiplication by a positive number other than 1 changes the magnitude of the vector, but not its direction. Scalar multiplication by β1 reverses it direction but doesnβt change its magnitude. Scalar multiplication by any negative number other than 1 both reverses the direction of the vector and changes it magnitude. The Distributive Property of Vector Addition is represented by the expression: π π΄ + π΅ = ππ΄ + ππ΅ where π is a scalar value
CONCEPT EXAMPLE: Perform the following vector operation of multiplying vector M by the scalar 3: π = 7π + 3π A. 9π + 21π B. 21π + 9π C. 21π + 3π D. 7π + 9π
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SOLUTION: The 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. The goal is to scale-up vector π(πβ) by the scalar 3. We can determine this using the concept of Multiplication of Vectors by a Scalar. When you multiply a vector by a scalar it is called "scaling" a vector, because you change how big or small the vector is. The numbers are called "scalars", because they "scale" the vector up or down. The rectangular form vector π΄ = π₯π + π¦π multiplied by the scalar a is ππ΄ = ππ₯π + ππ¦π In this problem, we are given: ππππππ β π’π π£πππ‘ππ π = πβ = π‘βπ π’πππππ€π π£πππ’π π€π πππ π πππ£πππ π = 7π + 3π π = 3 Plugging these values in to the multiplication of vectors by a scalar representation, we get: aM = M ] = axi + ayj
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3π = π] = 3 7 π + 3 3 π π] = 21π + 9π It is important to make sure that you distribute the scalar value to both of the π πππ π components. In this case, the scalar is only multiplied to the π component.
Therefore, the correct answer choice is B. πππ + ππ
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