Scalar Product: Essential Concepts for Competitive Exams

Scalar Product

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

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

Solution

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

Correct Answer: B — 0

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Q. If the vectors A = 6i + 8j and B = 2i + 3j, what is A · B? (2020)

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

Solution

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

Correct Answer: A — 42

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Q. If the vectors A = 6i + 8j and B = 3i + 4j are perpendicular, what is the value of A · B? (2021)

A. 0
B. 18
C. 30
D. 42

Solution

A · B = (6)(3) + (8)(4) = 18 + 32 = 50, but since they are perpendicular, A · B = 0.

Correct Answer: A — 0

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Q. If the vectors A and B are such that A · B = |A| |B|, what is the angle between them?

A. 0°
B. 90°
C. 180°
D. None of the above

Solution

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

Correct Answer: A — 0°

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Q. The scalar product of two unit vectors is 0. What can be inferred 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 two unit vectors is 0.5. What is the angle between them?

A. 60°
B. 30°
C. 90°
D. 120°

Solution

cos(θ) = 0.5, θ = cos⁻¹(0.5) = 60°.

Correct Answer: A — 60°

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Q. The scalar product of two vectors A and B is 12, and the angle between them is 60°. If |A| = 4, find |B|.

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

Solution

A · B = |A||B|cos(θ) => 12 = 4|B|(0.5) => |B| = 6.

Correct Answer: B — 8

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Q. What is the scalar product of A = 1i + 2j + 3k and B = 4i + 5j + 6k?

A. 32
B. 34
C. 36
D. 30

Solution

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

Correct Answer: A — 32

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Q. What is the scalar product of the vectors A = 1i + 2j + 3k and B = 4i + 5j + 6k?

A. 32
B. 34
C. 36
D. 30

Solution

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

Correct Answer: A — 32

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Scalar Product

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