Scalar Product: Essential Concepts for Competitive Exams

Scalar Product

Q. If A = 5i + 5j and B = 5i - 5j, what is the value of A · B?

A. 0
B. 25
C. 50
D. 10

Solution

A · B = (5)(5) + (5)(-5) = 25 - 25 = 0.

Correct Answer: B — 25

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Q. If A = 5i - 2j + k and B = 3i + 4j - 2k, calculate A · B.

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

Solution

A · B = (5)(3) + (-2)(4) + (1)(-2) = 15 - 8 - 2 = 5.

Correct Answer: A — -1

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Q. If A = 5i - 2j and B = -3i + 4j, calculate A · B. (2023)

A. -23
B. -7
C. 7
D. 23

Solution

A · B = (5)(-3) + (-2)(4) = -15 - 8 = -23

Correct Answer: A — -23

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Q. If A = 5i - 2j and B = -3i + 4j, what is the value of A · B?

A. -23
B. -7
C. 7
D. 23

Solution

A · B = (5)(-3) + (-2)(4) = -15 - 8 = -23.

Correct Answer: A — -23

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Q. If A = 6i + 8j and B = 2i + 3j, calculate A · B.

A. 36
B. 42
C. 38
D. 30

Solution

A · B = (6)(2) + (8)(3) = 12 + 24 = 36.

Correct Answer: A — 36

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Q. If A = 6i + 8j and B = 2i + 3j, find A · B.

A. 36
B. 42
C. 30
D. 24

Solution

A · B = (6)(2) + (8)(3) = 12 + 24 = 36.

Correct Answer: A — 36

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Q. If A = 6i + 8j and B = 2i + 3j, find the scalar product A · B.

A. 42
B. 54
C. 48
D. 36

Solution

A · B = (6)(2) + (8)(3) = 12 + 24 = 36.

Correct Answer: C — 48

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Q. If A = 6i + 8j and B = 2i + 3j, what is the scalar product A · B?

A. 42
B. 36
C. 30
D. 48

Solution

A · B = (6)(2) + (8)(3) = 12 + 24 = 36.

Correct Answer: B — 36

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Q. If A = 6i + 8j and B = 2i + 3j, what is the value of A · B?

A. 54
B. 60
C. 48
D. 42

Solution

A · B = (6)(2) + (8)(3) = 12 + 24 = 36.

Correct Answer: A — 54

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Q. If A = 6i + 8j and B = 3i + 4j, what is the angle between A and B?

A. 0°
B. 30°
C. 45°
D. 90°

Solution

A · B = 6*3 + 8*4 = 18 + 32 = 50. |A| = 10, |B| = 5. cos(θ) = 50/(10*5) = 1. θ = 0°.

Correct Answer: D — 90°

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Q. If A = 6i + 8j and B = 3i + 4j, what is the value of A · B?

A. 50
B. 54
C. 48
D. 52

Solution

A · B = (6)(3) + (8)(4) = 18 + 32 = 50.

Correct Answer: C — 48

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Q. If A = 6i - 2j and B = -3i + 4j, find A · B.

A. -18
B. -24
C. -12
D. -6

Solution

A · B = (6)(-3) + (-2)(4) = -18 - 8 = -26.

Correct Answer: A — -18

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Q. If A = 7i + 1j and B = 1i + 7j, calculate A · B.

A. 56
B. 58
C. 60
D. 62

Solution

A · B = (7)(1) + (1)(7) = 7 + 7 = 14.

Correct Answer: A — 56

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Q. If A = 7i + 1j and B = 1i + 7j, find A · B.

A. 56
B. 14
C. 8
D. 50

Solution

A · B = (7)(1) + (1)(7) = 7 + 7 = 14.

Correct Answer: A — 56

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Q. If A = 7i + 1j and B = 2i + 3j, what is the scalar product A · B? (2022)

A. 23
B. 21
C. 19
D. 17

Solution

A · B = (7)(2) + (1)(3) = 14 + 3 = 17

Correct Answer: A — 23

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Q. If A = 7i - 2j and B = -3i + 4j, find A · B. (2021)

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

Solution

A · B = (7)(-3) + (-2)(4) = -21 - 8 = -29.

Correct Answer: A — -26

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Q. If A = 7i - 3j and B = -2i + 5j, what is the value of A · B?

A. -29
B. -31
C. -25
D. -27

Solution

A · B = (7)(-2) + (-3)(5) = -14 - 15 = -29.

Correct Answer: A — -29

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Q. If A = ai + bj and B = ci + dj, what is the expression for A · B?

A. ac + bd
B. ab + cd
C. ad + bc
D. a + b + c + d

Solution

A · B = (ai + bj) · (ci + dj) = ac + bd.

Correct Answer: A — ac + bd

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Q. If A = i + 2j + 3k and B = 4i + 5j + 6k, what is the value of A · B?

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. If the angle between two vectors A and B is 60 degrees and |A| = 5, |B| = 10, what is the scalar product A · B? (2020)

A. 25
B. 30
C. 35
D. 50

Solution

A · B = |A||B|cos(60°) = 5 * 10 * 0.5 = 25

Correct Answer: B — 30

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Q. If the angle between two vectors A and B is 60 degrees and |A| = 5, |B| = 10, what is A · B?

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

Solution

A · B = |A||B|cos(60°) = 5 * 10 * 0.5 = 25.

Correct Answer: B — 30

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

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

Solution

A · B = |A||B|cos(90°) = |A||B| * 0 = 0.

Correct Answer: A — 0

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

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

Solution

If A · B = 0, then A and B are orthogonal (perpendicular) to each other.

Correct Answer: C — They are orthogonal

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Q. If the scalar product of two vectors A and B is 15 and the magnitudes are |A| = 5 and |B| = 3, find the angle between them.

A. 60°
B. 45°
C. 30°
D. 90°

Solution

A · B = |A||B|cos(θ) => 15 = 5*3*cos(θ) => cos(θ) = 1, θ = 0°.

Correct Answer: B — 45°

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Q. If the scalar product of two vectors A and B is equal to the product of their magnitudes, what can be inferred?

A. They are perpendicular
B. They are parallel
C. They are equal
D. They are opposite

Solution

If A · B = |A||B|, then the angle between them is 0°, meaning they are parallel.

Correct Answer: B — They are parallel

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Q. If the scalar product of vectors A and B is equal to the product of their magnitudes, what can be said about the angle between them?

A. 0°
B. 90°
C. 180°
D. 45°

Solution

If A · B = |A||B|, then cos(θ) = 1, which means θ = 0°.

Correct Answer: A — 0°

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Q. If the vectors A = 1i + 2j and B = 2i + 1j, find A · B. (2023)

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

Solution

A · B = (1)(2) + (2)(1) = 2 + 2 = 4

Correct Answer: B — 5

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Q. If the vectors A = 2i + 2j and B = 2i - 2j, find A · B.

A. 0
B. 4
C. 8
D. 2

Solution

A · B = (2)(2) + (2)(-2) = 4 - 4 = 0.

Correct Answer: B — 4

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Q. If the vectors A = 2i + 3j and B = 3i + 4j are perpendicular, what is the value of A · B?

A. 0
B. 6
C. 12
D. 9

Solution

Since A and B are perpendicular, A · B = 0.

Correct Answer: A — 0

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

A. 25
B. 30
C. 32
D. 28

Solution

A · B = (3)(4) + (4)(3) = 12 + 12 = 24.

Correct Answer: A — 25

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Scalar Product

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