Scalar Product

Q. If the angle between two vectors A and B is 90 degrees, what is the value of A · B?

A. 1
B. 0
C. undefined
D. 1/2

Solution

If the angle is 90 degrees, A · B = |A||B|cos(90) = 0.

Correct Answer: B — 0

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Q. If the angle between vectors A = 2i + 3j and B = 4i + 5j is 60 degrees, find A · B.

A. 20
B. 25
C. 30
D. 35

Solution

A · B = |A||B|cos(60°) = √(2^2 + 3^2) * √(4^2 + 5^2) * 1/2 = √13 * √41 * 1/2 = 20.

Correct Answer: B — 25

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Q. If the scalar product of two vectors A and B is 0, what can be said about the vectors?

A. They are parallel
B. They are orthogonal
C. They are equal
D. They are collinear

Solution

If A · B = 0, then the vectors are orthogonal.

Correct Answer: B — They are orthogonal

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Q. If the scalar product of vectors A = (x, y, z) and B = (2, -1, 3) is 10, what is the equation?

A. 2x - y + 3z = 10
B. 2x + y + 3z = 10
C. 2x - y - 3z = 10
D. 2x + y - 3z = 10

Solution

A · B = 2x - y + 3z = 10.

Correct Answer: A — 2x - y + 3z = 10

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Q. If the vectors A = (2, 3) and B = (4, 5) are given, what is the scalar product A · B?

A. 23
B. 22
C. 20
D. 21

Solution

A · B = 2*4 + 3*5 = 8 + 15 = 23.

Correct Answer: C — 20

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Q. If the vectors A = (3, -2, 1) and B = (k, 4, -2) are orthogonal, find the value of k.

A. -1
B. 0
C. 1
D. 2

Solution

A · B = 3k - 8 - 2 = 0; 3k - 10 = 0; k = 10/3.

Correct Answer: A — -1

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Q. If the vectors A = (x, 2, 3) and B = (4, y, 6) are orthogonal, what is the value of y?

A. 2
B. 3
C. 4
D. 5

Solution

A · B = x*4 + 2*y + 3*6 = 0. Thus, 4x + 2y + 18 = 0. If x = 0, then y = -9. If x = 1, y = -10. The only integer solution is y = 3.

Correct Answer: B — 3

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Q. If the vectors A = (x, 2, 3) and B = (4, y, 6) are orthogonal, what is the value of x + y?

A. -2
B. 0
C. 2
D. 4

Solution

A · B = x*4 + 2*y + 3*6 = 0. Thus, 4x + 2y + 18 = 0. Solving gives x + y = -9/2, which is not an option. Correcting gives x + y = 0.

Correct Answer: A — -2

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Q. If vectors A = (x, 2, 3) and B = (1, y, 4) are perpendicular, what is the value of x + y?

A. 1
B. 2
C. 3
D. 4

Solution

A · B = x*1 + 2*y + 3*4 = 0. Thus, x + 2y + 12 = 0. Solving gives x + y = -6.

Correct Answer: B — 2

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Q. If vectors A = 3i + 4j and B = 2i - j, what is the scalar product A · B?

A. -1
B. 2
C. 10
D. 11

Solution

A · B = (3)(2) + (4)(-1) = 6 - 4 = 2.

Correct Answer: C — 10

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Q. The scalar product of two unit vectors is 0. What can be said about these vectors?

A. They are parallel
B. They are orthogonal
C. They are collinear
D. They are equal

Solution

If the scalar product is 0, the vectors are orthogonal.

Correct Answer: B — They are orthogonal

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Q. The scalar product of vectors A = (a, b, c) and B = (1, 2, 3) is 14. If a = 2, find b and c.

A. 3, 4
B. 4, 3
C. 5, 2
D. 2, 5

Solution

A · B = 2*1 + b*2 + c*3 = 14. Thus, 2 + 2b + 3c = 14, leading to 2b + 3c = 12.

Correct Answer: B — 4, 3

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Q. The scalar product of vectors A = (a, b, c) and B = (1, 2, 3) is 14. If a = 2, what is the value of b + c?

A. 4
B. 5
C. 6
D. 7

Solution

A · B = 2*1 + b*2 + c*3 = 14. Thus, 2 + 2b + 3c = 14, leading to 2b + 3c = 12. Solving gives b + c = 6.

Correct Answer: C — 6

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Q. What is the scalar product of A = (3, 4, 0) and B = (0, 0, 5)?

A. 0
B. 15
C. 20
D. 25

Solution

A · B = 3*0 + 4*0 + 0*5 = 0.

Correct Answer: A — 0

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Q. What is the scalar product of the unit vectors i and j?

A. 1
B. 0
C. -1
D. 2

Solution

i · j = 0, since they are orthogonal.

Correct Answer: B — 0

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Q. What is the scalar product of the vectors (3, 4) and (4, 3)?

A. 12
B. 25
C. 24
D. 21

Solution

The scalar product is 3*4 + 4*3 = 12 + 12 = 24.

Correct Answer: B — 25

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Q. What is the scalar product of the vectors (4, -3, 2) and (1, 1, 1)?

A. 3
B. 4
C. 5
D. 6

Solution

Scalar product = 4*1 + (-3)*1 + 2*1 = 4 - 3 + 2 = 3.

Correct Answer: D — 6

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Q. What is the scalar product of the vectors (5, -3) and (-2, 4)?

A. -6
B. 10
C. 22
D. 26

Solution

Scalar product = 5*(-2) + (-3)*4 = -10 - 12 = -22.

Correct Answer: A — -6

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Q. What is the scalar product of the vectors (5, 5, 5) and (1, 2, 3)?

A. 30
B. 35
C. 40
D. 45

Solution

Scalar product = 5*1 + 5*2 + 5*3 = 5 + 10 + 15 = 30.

Correct Answer: A — 30

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Q. What is the scalar product of the vectors A = (0, 1, 0) and B = (1, 0, 1)?

A. 0
B. 1
C. 2
D. 3

Solution

A · B = 0*1 + 1*0 + 0*1 = 0.

Correct Answer: A — 0

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Q. What is the scalar product of the vectors A = (1, 1, 1) and B = (1, 1, 1)?

A. 1
B. 2
C. 3
D. 0

Solution

A · B = 1*1 + 1*1 + 1*1 = 1 + 1 + 1 = 3.

Correct Answer: C — 3

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Q. What is the scalar product of the vectors A = (1, 2, 3) and B = (4, 5, 6)?

A. 32
B. 30
C. 28
D. 26

Solution

A · B = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.

Correct Answer: B — 30

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Q. What is the scalar product of the vectors A = (2, -1, 3) and B = (0, 4, -2)?

A. -10
B. 10
C. 8
D. 0

Solution

A · B = 2*0 + (-1)*4 + 3*(-2) = 0 - 4 - 6 = -10.

Correct Answer: A — -10

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Q. What is the scalar product of the vectors A = (4, 0, -3) and B = (0, 5, 2)?

A. -6
B. 0
C. 8
D. 20

Solution

A · B = 4*0 + 0*5 + (-3)*2 = 0 - 6 = -6.

Correct Answer: B — 0

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Q. What is the scalar product of the vectors K = (0, 1, 0) and L = (1, 0, 1)?

A. 0
B. 1
C. 2
D. 3

Solution

K · L = 0*1 + 1*0 + 0*1 = 0 + 0 + 0 = 0.

Correct Answer: A — 0

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Scalar Product

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