83.1 Dot Product Problem Set
DOT PRODUCT | PRACTICE PROBLEMS Complete the following to reinforce your understanding of the concept covered in this module.
PROBLEM 1: The dot product of the two vectors π =< 0,0,3 > πππ π =< 2,0,2 > with an angle of 45Β° between them, is most close to: A. 6 B. 5 C. 9 D. 24
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PROBLEM 2: The dot product of two vectors π =< 0,2, β1 > and π =< β1,1,2 > with an angle of 90Β° between them, is most close to: A. β9 B. 9 C. 81 D. 0
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PROBLEM 3: The angle between the two vectors π =< 2,5, 0 > and π =< β3,2, 0 > is most close to: A. 77.2 B. 43.8 C. 78.1 D. 5
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DOT PRODUCT | SOLUTIONS SOLUTION 1: The DOT PRODUCT is one of two OPERATIONS that can be deployed using VECTOR MULTIPLICATION, the other operation being the CROSS PRODUCT. The GENERAL FORMULA for the DOT PRODUCT 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 π΄ =< π! , π! , π! > and π΅ =< π! , π! , π! > with an angle π between them, where π is between 0 and π, the DOT PRODUCT can be determined using the following formula: π΄ β π΅ = π! π! + π! π! + π! π! = π΄ π΅ cos π = π΅ β π΄ The difference between the DOT PRODUCT and the CROSS PRODUCT is that the result of a DOT PRODUCT is a SINGLE SCALAR VALUE, which is why it is sometimes referred to as the SCALAR PRODUCT. In this problem, we are given the two VECTORS: π =< 0,0,3 > π =< 2,0,2 >
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As well as an ANGLE: π = 45Β° Revisiting this formula, we find that the DOT PRODUCT can quantified using a relationship between two VECTORS and the ANGLE between those VECTORS, such that: π΄ β π΅ = π΄ π΅ cos π We have all the foundational data we need, but before we move forward with our calculations, we do need to determine the MAGNITUDE of each VECTOR. The MAGNITUDE of a VECTOR, can be determined using the formula:
π΄ =
π! ! + π! ! + π! !
Establishing the MAGNITUDE of VECTOR A, we get:
π΄ =
(0)! + (0)! + (3)! = 3
Establishing the MAGNITUDE of VECTOR B, we get:
π΄ =
(2)! + (0)! + (2)! = 2 2
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Knowing that the angle is 45Β°, we can plug in the values to determine the DOT PRODUCT, which gives us π΄ β π΅ = 3 β 2 2 cos 45 Or: π΄βπ΅ =6 The correct answer choice is A. π
SOLUTION 2: The GENERAL FORMULA for the DOT PRODUCT 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 π΄ =< π! , π! , π! > and π΅ =< π! , π! , π! > with an angle π between them, where π is between 0 and π, the DOT PRODUCT can be determined using the following formula: π΄ β π΅ = π! π! + π! π! + π! π! = π΄ π΅ cos π = π΅ β π΄ The difference between the DOT PRODUCT and the CROSS PRODUCT is that the result of a DOT PRODUCT is a SINGLE SCALAR VALUE, which is why it is sometimes referred to as the SCALAR PRODUCT.
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In this problem, we are given the two VECTORS: π =< 0,2, β1 > π =< β1,1,2 > As well as an ANGLE: π = 90Β° Revisiting this formula, we find that the DOT PRODUCT can quantified using a relationship between two VECTORS and the ANGLE between those VECTORS, such that: π΄ β π΅ = π΄ π΅ cos π From here, there are a couple routes people will take. One subset will move forward with determining the MAGNITUDE of both VECTOR A and VECTOR B and then continue to plug everything in to define the DOT PRODUCT. A second subset, one in which I hope you will fall in to, will realize that with an angle of 90Β° we are dealing with a special case of the DOT PRODUCTβ¦one where the two vectors are orthogonal, or perpendicular with one another. The FORMULAS for the DOT PRODUCT of TWO ORTHOGONAL VECTORS can be referenced under the topic of MATHEMATICS on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
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Since the cosine of 90Β° is zero, the DOT PRODUCT of two orthogonal vectors will result in zero as well. We could go through all the work to show this, but this is really what it comes down to: cos 90 = 0 Which means: π΄βπ΅ = π΄ π΅ 0 =0 It doesnβt matter what the VECTORS look like, the end result is always the same for TWO ORTHOGONAL VECTORS. The correct answer choice is D. π
SOLUTION 3: The DOT PRODUCT is one of two OPERATIONS that can be deployed using VECTOR MULTIPLICATION, the other operation being the CROSS PRODUCT. The GENERAL FORMULA for the DOT PRODUCT can be referenced under the subject of MATHEMATICS on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
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Given two vectors π΄ =< π! , π! , π! > and π΅ =< π! , π! , π! > with an angle π between them, where π is between 0 and π, the DOT PRODUCT can be determined using the following formula: π΄ β π΅ = π! π! + π! π! + π! π! = π΄ π΅ cos π = π΅ β π΄ The difference between the DOT PRODUCT and the CROSS PRODUCT is that the result of a DOT PRODUCT is a SINGLE SCALAR VALUE, which is why it is sometimes referred to as the SCALAR PRODUCT. In this problem, we are given the two VECTORS: π =< 2,5, 0 > π =< β3,2, 0 > And we are asked to determine the ANGLE between the two. Revisiting our GENERAL FORMULA, we find that the DOT PRODUCT can quantified using a relationship between two VECTORS and the ANGLE between those VECTORS, such that: π΄ β π΅ = π΄ π΅ cos π We also know that the DOT PRODUCT can be quantified using the formula: π΄ β π΅ = π! π! + π! π! + π! π!
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Which means that we will lean on the relationship: π! π! + π! π! + π! π! = π΄ π΅ cos π We have all the foundational data we need to solve this problem, but before we move forward with our calculations, we do need to determine the MAGNITUDE of each VECTOR. The MAGNITUDE of a VECTOR, can be determined using the formula:
π΄ =
π! ! + π! ! + π! !
Establishing the MAGNITUDE of VECTOR A, we get:
π΄ =
(2)! + (5)! + (0)! = 29
Establishing the MAGNITUDE of VECTOR B, we get:
π΄ =
(β3)! + (2)! + (0)! = 13
Plugging in all of the data points we have defined up to this point, we get: 2 β3 + 5 2 + 0 0 = 29 13 cos π Rearranging to solve for the ANGLE we get:
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π = cos !!
4 29 13
= 78.1Β°
The correct answer choice is C. ππ. π
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PROBLEM 1: The dot product of the two vectors π =< 0,0,3 > πππ π =< 2,0,2 > with an angle of 45Β° between them, is most close to: A. 6 B. 5 C. 9 D. 24
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PROBLEM 2: The dot product of two vectors π =< 0,2, β1 > and π =< β1,1,2 > with an angle of 90Β° between them, is most close to: A. β9 B. 9 C. 81 D. 0
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PROBLEM 3: The angle between the two vectors π =< 2,5, 0 > and π =< β3,2, 0 > is most close to: A. 77.2 B. 43.8 C. 78.1 D. 5
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DOT PRODUCT | SOLUTIONS SOLUTION 1: The DOT PRODUCT is one of two OPERATIONS that can be deployed using VECTOR MULTIPLICATION, the other operation being the CROSS PRODUCT. The GENERAL FORMULA for the DOT PRODUCT 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 π΄ =< π! , π! , π! > and π΅ =< π! , π! , π! > with an angle π between them, where π is between 0 and π, the DOT PRODUCT can be determined using the following formula: π΄ β π΅ = π! π! + π! π! + π! π! = π΄ π΅ cos π = π΅ β π΄ The difference between the DOT PRODUCT and the CROSS PRODUCT is that the result of a DOT PRODUCT is a SINGLE SCALAR VALUE, which is why it is sometimes referred to as the SCALAR PRODUCT. In this problem, we are given the two VECTORS: π =< 0,0,3 > π =< 2,0,2 >
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As well as an ANGLE: π = 45Β° Revisiting this formula, we find that the DOT PRODUCT can quantified using a relationship between two VECTORS and the ANGLE between those VECTORS, such that: π΄ β π΅ = π΄ π΅ cos π We have all the foundational data we need, but before we move forward with our calculations, we do need to determine the MAGNITUDE of each VECTOR. The MAGNITUDE of a VECTOR, can be determined using the formula:
π΄ =
π! ! + π! ! + π! !
Establishing the MAGNITUDE of VECTOR A, we get:
π΄ =
(0)! + (0)! + (3)! = 3
Establishing the MAGNITUDE of VECTOR B, we get:
π΄ =
(2)! + (0)! + (2)! = 2 2
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Knowing that the angle is 45Β°, we can plug in the values to determine the DOT PRODUCT, which gives us π΄ β π΅ = 3 β 2 2 cos 45 Or: π΄βπ΅ =6 The correct answer choice is A. π
SOLUTION 2: The GENERAL FORMULA for the DOT PRODUCT 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 π΄ =< π! , π! , π! > and π΅ =< π! , π! , π! > with an angle π between them, where π is between 0 and π, the DOT PRODUCT can be determined using the following formula: π΄ β π΅ = π! π! + π! π! + π! π! = π΄ π΅ cos π = π΅ β π΄ The difference between the DOT PRODUCT and the CROSS PRODUCT is that the result of a DOT PRODUCT is a SINGLE SCALAR VALUE, which is why it is sometimes referred to as the SCALAR PRODUCT.
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In this problem, we are given the two VECTORS: π =< 0,2, β1 > π =< β1,1,2 > As well as an ANGLE: π = 90Β° Revisiting this formula, we find that the DOT PRODUCT can quantified using a relationship between two VECTORS and the ANGLE between those VECTORS, such that: π΄ β π΅ = π΄ π΅ cos π From here, there are a couple routes people will take. One subset will move forward with determining the MAGNITUDE of both VECTOR A and VECTOR B and then continue to plug everything in to define the DOT PRODUCT. A second subset, one in which I hope you will fall in to, will realize that with an angle of 90Β° we are dealing with a special case of the DOT PRODUCTβ¦one where the two vectors are orthogonal, or perpendicular with one another. The FORMULAS for the DOT PRODUCT of TWO ORTHOGONAL VECTORS can be referenced under the topic of MATHEMATICS on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
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Since the cosine of 90Β° is zero, the DOT PRODUCT of two orthogonal vectors will result in zero as well. We could go through all the work to show this, but this is really what it comes down to: cos 90 = 0 Which means: π΄βπ΅ = π΄ π΅ 0 =0 It doesnβt matter what the VECTORS look like, the end result is always the same for TWO ORTHOGONAL VECTORS. The correct answer choice is D. π
SOLUTION 3: The DOT PRODUCT is one of two OPERATIONS that can be deployed using VECTOR MULTIPLICATION, the other operation being the CROSS PRODUCT. The GENERAL FORMULA for the DOT PRODUCT can be referenced under the subject of MATHEMATICS on page 35 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
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Given two vectors π΄ =< π! , π! , π! > and π΅ =< π! , π! , π! > with an angle π between them, where π is between 0 and π, the DOT PRODUCT can be determined using the following formula: π΄ β π΅ = π! π! + π! π! + π! π! = π΄ π΅ cos π = π΅ β π΄ The difference between the DOT PRODUCT and the CROSS PRODUCT is that the result of a DOT PRODUCT is a SINGLE SCALAR VALUE, which is why it is sometimes referred to as the SCALAR PRODUCT. In this problem, we are given the two VECTORS: π =< 2,5, 0 > π =< β3,2, 0 > And we are asked to determine the ANGLE between the two. Revisiting our GENERAL FORMULA, we find that the DOT PRODUCT can quantified using a relationship between two VECTORS and the ANGLE between those VECTORS, such that: π΄ β π΅ = π΄ π΅ cos π We also know that the DOT PRODUCT can be quantified using the formula: π΄ β π΅ = π! π! + π! π! + π! π!
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Which means that we will lean on the relationship: π! π! + π! π! + π! π! = π΄ π΅ cos π We have all the foundational data we need to solve this problem, but before we move forward with our calculations, we do need to determine the MAGNITUDE of each VECTOR. The MAGNITUDE of a VECTOR, can be determined using the formula:
π΄ =
π! ! + π! ! + π! !
Establishing the MAGNITUDE of VECTOR A, we get:
π΄ =
(2)! + (5)! + (0)! = 29
Establishing the MAGNITUDE of VECTOR B, we get:
π΄ =
(β3)! + (2)! + (0)! = 13
Plugging in all of the data points we have defined up to this point, we get: 2 β3 + 5 2 + 0 0 = 29 13 cos π Rearranging to solve for the ANGLE we get:
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π = cos !!
4 29 13
= 78.1Β°
The correct answer choice is C. ππ. π
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