Scalar Product: Essential Concepts for Competitive Exams

Scalar Product

Q. If A = 2i + 2j and B = 3i + 4j, what is the angle between A and B?

A. 45 degrees
B. 60 degrees
C. 30 degrees
D. 90 degrees

Solution

cos(θ) = (A · B) / (|A||B|) = (2*3 + 2*4) / (√(2^2 + 2^2) * √(3^2 + 4^2)) = 0.5, θ = 60 degrees.

Correct Answer: B — 60 degrees

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Q. If A = 2i + 3j + k and B = i - 2j + 4k, find the scalar product A · B.

A. 10
B. 12
C. 14
D. 8

Solution

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

Correct Answer: B — 12

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Q. If A = 2i + 3j and B = 3i + 4j, find the angle between A and B.

A. 45 degrees
B. 60 degrees
C. 30 degrees
D. 90 degrees

Solution

cos(θ) = (A · B) / (|A||B|) = (2*3 + 3*4) / (√(2^2 + 3^2) * √(3^2 + 4^2)) = 18 / (√13 * 5) = 0.6, θ = 60 degrees.

Correct Answer: B — 60 degrees

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

A. 18
B. 20
C. 22
D. 16

Solution

A · B = (2)(3) + (3)(4) = 6 + 12 = 18.

Correct Answer: A — 18

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Q. If A = 2i + 3j and B = 4i + 0j, what is the value of A · B? (2022)

A. 6
B. 8
C. 12
D. 14

Solution

A · B = (2)(4) + (3)(0) = 8 + 0 = 8

Correct Answer: B — 8

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Q. If A = 2i + 3j and B = 4i + 5j, find the angle θ between A and B if A · B = |A||B|cos(θ).

A. 0°
B. 45°
C. 60°
D. 90°

Solution

A · B = 2*4 + 3*5 = 8 + 15 = 23. |A| = √(2^2 + 3^2) = √13, |B| = √(4^2 + 5^2) = √41. cos(θ) = 23 / (√13 * √41).

Correct Answer: C — 60°

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

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

Solution

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

Correct Answer: A — 23

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

A. 14
B. 8
C. 10
D. 12

Solution

A · B = (2)(4) + (3)(0) + (0)(1) = 8.

Correct Answer: A — 14

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Q. If A = 2i + 3j and B = ai + bj, and A · B = 0, what is the relationship between a and b?

A. a = 3b/2
B. a = -3b/2
C. a = 2b/3
D. a = -2b/3

Solution

A · B = 2a + 3b = 0 implies a = -3b/2.

Correct Answer: B — a = -3b/2

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Q. If A = 3i + 4j and B = 0i + 0j, what is A · B? (2023)

A. 0
B. 3
C. 4
D. 7

Solution

A · B = (3)(0) + (4)(0) = 0.

Correct Answer: A — 0

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Q. If A = 3i + 4j and B = 0i + 0j, what is the scalar product A · B?

A. 0
B. 12
C. 7
D. 3

Solution

A · B = (3)(0) + (4)(0) = 0.

Correct Answer: A — 0

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

A. 25
B. 24
C. 26
D. 22

Solution

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

Correct Answer: A — 25

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Q. If A = 3i + 4j and B = 4i + 3j, find the angle between A and B.

A. 45°
B. 60°
C. 90°
D. 30°

Solution

A · B = 3*4 + 4*3 = 12 + 12 = 24. |A| = 5, |B| = 5. cos(θ) = 24 / (5*5) = 24/25. θ = cos^(-1)(24/25) ≈ 60°.

Correct Answer: B — 60°

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

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

Solution

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

Correct Answer: B — 30

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

A. -11
B. -5
C. -7
D. -9

Solution

A · B = (3)(-1) + (-2)(4) = -3 - 8 = -11.

Correct Answer: A — -11

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Q. If A = 4i + 3j and B = 0i + 0j, what is A · B?

A. 0
B. 12
C. 7
D. 4

Solution

A · B = (4)(0) + (3)(0) = 0.

Correct Answer: A — 0

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Q. If A = 4i + 3j and B = 0i + 5j, what is A · B? (2019)

A. 0
B. 15
C. 20
D. 12

Solution

A · B = (4)(0) + (3)(5) = 0 + 15 = 15

Correct Answer: B — 15

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

A. 25
B. 24
C. 27
D. 20

Solution

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

Correct Answer: A — 25

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Q. If A = 4i + 3j and B = 4i - 3j, what is A · B?

A. 16
B. 0
C. 25
D. 12

Solution

A · B = (4)(4) + (3)(-3) = 16 - 9 = 7.

Correct Answer: A — 16

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Q. If A = 4i + 3j and B = ai + bj, and A · B = 24, find the value of a + b if a = 2.

A. 10
B. 12
C. 8
D. 6

Solution

A · B = (4)(2) + (3)(b) = 8 + 3b = 24, 3b = 16, b = 16/3. a + b = 2 + 16/3 = 10.

Correct Answer: B — 12

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Q. If A = 4i + 5j and B = 1i + 2j, calculate the scalar product A · B. (2022)

A. 14
B. 15
C. 16
D. 17

Solution

A · B = (4)(1) + (5)(2) = 4 + 10 = 14.

Correct Answer: A — 14

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

A. -7
B. 7
C. 1
D. 0

Solution

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

Correct Answer: A — -7

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

A. -7
B. 7
C. 1
D. 0

Solution

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

Correct Answer: A — -7

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Q. If A = 5i + 12j and B = 12i - 5j, what is the scalar product A · B?

A. 0
B. 60
C. 70
D. 80

Solution

A · B = (5)(12) + (12)(-5) = 60 - 60 = 0.

Correct Answer: B — 60

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Q. If A = 5i + 12j and B = 3i + 4j, find A · B.

A. 63
B. 60
C. 64
D. 65

Solution

A · B = (5)(3) + (12)(4) = 15 + 48 = 63.

Correct Answer: A — 63

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Q. If A = 5i + 2j + 3k and B = 1i - 4j + 2k, find A · B.

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

Solution

A · B = (5)(1) + (2)(-4) + (3)(2) = 5 - 8 + 6 = 3.

Correct Answer: B — 0

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Q. If A = 5i + 2j + 3k and B = 4i - j + 2k, find the scalar product A · B.

A. 26
B. 28
C. 30
D. 24

Solution

A · B = (5)(4) + (2)(-1) + (3)(2) = 20 - 2 + 6 = 24.

Correct Answer: B — 28

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Q. If A = 5i + 2j + 3k and B = 4i - j + 6k, find A · B. (2020)

A. 38
B. 32
C. 30
D. 28

Solution

A · B = (5)(4) + (2)(-1) + (3)(6) = 20 - 2 + 18 = 36.

Correct Answer: B — 32

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Q. If A = 5i + 2j and B = 3i + 6j, find the angle θ between A and B if A · B = |A||B|cos(θ).

A. 60°
B. 45°
C. 30°
D. 90°

Solution

A · B = 5*3 + 2*6 = 15 + 12 = 27. |A| = √(5^2 + 2^2) = √29, |B| = √(3^2 + 6^2) = √45. cos(θ) = 27/(√29 * √45). θ = 45°.

Correct Answer: B — 45°

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

A. -1
B. 7
C. 1
D. 0

Solution

A · B = (5)(3) + (2)(-4) = 15 - 8 = 7.

Correct Answer: A — -1

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Scalar Product

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