Vector Algebra for Competitive Exam Success

Vector Algebra

Scalar Product Vector Product Vectors in 2D & 3D

Q. Find the angle between the vectors A = 2i + 2j and B = 2i - 2j. (2022)

A. 0 degrees
B. 45 degrees
C. 90 degrees
D. 180 degrees

Solution

cos(θ) = (A · B) / (|A||B|). A · B = 0, hence θ = 90 degrees.

Correct Answer: C — 90 degrees

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Q. Find the angle between the vectors A = i + j and B = 2i + 2j.

A. 0 degrees
B. 45 degrees
C. 90 degrees
D. 60 degrees

Solution

cos(θ) = (A · B) / (|A| |B|). A · B = 1*2 + 1*2 = 4; |A| = √2, |B| = 2√2. Thus, cos(θ) = 4 / (√2 * 2√2) = 1, θ = 0 degrees.

Correct Answer: A — 0 degrees

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Q. Find the angle between the vectors A = i + j and B = i - j.

A. 0 degrees
B. 45 degrees
C. 90 degrees
D. 135 degrees

Solution

cos(θ) = (A · B) / (|A||B|) = (1 - 1) / (√2 * √2) = 0, θ = 90 degrees.

Correct Answer: C — 90 degrees

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Q. Find the angle between the vectors A = i + j and B = j - i. (2022)

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

Solution

cos(θ) = (A · B) / (|A| |B|). A · B = 0, hence θ = 90 degrees.

Correct Answer: A — 90 degrees

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Q. Find the angle θ between the vectors A = i + 2j and B = 2i + 3j if A · B = |A||B|cos(θ).

A. 60°
B. 45°
C. 30°
D. 90°

Solution

A · B = 1*2 + 2*3 = 8. |A| = √(1^2 + 2^2) = √5, |B| = √(2^2 + 3^2) = √13. cos(θ) = 8/(√5*√13).

Correct Answer: B — 45°

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Q. Find the angle θ between vectors A = 4i + 3j and B = 1i + 2j if A · B = |A||B|cos(θ).

A. 60°
B. 45°
C. 30°
D. 90°

Solution

A · B = 4*1 + 3*2 = 10; |A| = √(4^2 + 3^2) = 5; |B| = √(1^2 + 2^2) = √5; cos(θ) = 10/(5√5) = 2/√5; θ = 45°.

Correct Answer: B — 45°

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Q. Find the scalar product of A = 2i + 3j + k and B = i + 2j + 3k. (2020)

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

Solution

A · B = (2)(1) + (3)(2) + (1)(3) = 2 + 6 + 3 = 11

Correct Answer: A — 14

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Q. Find the scalar product of A = 6i + 8j and B = 2i + 3j.

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

Solution

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

Correct Answer: A — 42

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Q. Find the scalar product of the vectors A = 7i - 2j + k and B = 3i + 4j - 5k.

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

Solution

A · B = (7)(3) + (-2)(4) + (1)(-5) = 21 - 8 - 5 = 8.

Correct Answer: A — -1

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Q. Find the scalar product of vectors A = 7i + 1j + 2k and B = 3i + 4j + 5k.

A. 43
B. 37
C. 35
D. 41

Solution

A · B = (7)(3) + (1)(4) + (2)(5) = 21 + 4 + 10 = 35.

Correct Answer: A — 43

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Q. Find the unit vector in the direction of vector A = 6i - 8j.

A. 3/5 i - 4/5 j
B. 6/10 i - 8/10 j
C. 1/5 i - 2/5 j
D. 2/5 i - 3/5 j

Solution

Magnitude |A| = √(6^2 + (-8)^2) = √(36 + 64) = 10. Unit vector = (6/10)i + (-8/10)j = (3/5)i - (4/5)j.

Correct Answer: A — 3/5 i - 4/5 j

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Q. Find the unit vector in the direction of vector D = -3i + 4j.

A. -0.6i + 0.8j
B. 0.6i - 0.8j
C. 0.8i + 0.6j
D. -0.8i + 0.6j

Solution

Magnitude |D| = √((-3)^2 + 4^2) = √(9 + 16) = √25 = 5. Unit vector = D/|D| = (-3/5)i + (4/5)j = -0.6i + 0.8j.

Correct Answer: A — -0.6i + 0.8j

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Q. For vectors A = 2i + 3j and B = 5i + 6j, what is A · B?

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

Solution

A · B = (2)(5) + (3)(6) = 10 + 18 = 28.

Correct Answer: A — 28

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

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

Solution

A · B = (2)(3) + (1)(4) = 6 + 4 = 10.

Correct Answer: A — 14

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Q. For vectors A = 4i + 3j and B = 3i - 4j, find A · B.

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

Solution

A · B = (4)(3) + (3)(-4) = 12 - 12 = 0.

Correct Answer: A — -6

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

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. Given A = 1i + 1j and B = 1i + 1j, what is A · B?

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

Solution

A · B = (1)(1) + (1)(1) = 1 + 1 = 2.

Correct Answer: A — 2

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

A. 12
B. 18
C. 10
D. 14

Solution

A · B = (2)(3) + (2)(3) = 6 + 6 = 12.

Correct Answer: A — 12

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

A. 14
B. 10
C. 12
D. 16

Solution

A · B = (2)(3) + (2)(4) = 6 + 8 = 14.

Correct Answer: A — 14

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

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

Solution

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

Correct Answer: C — 8

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Q. Given vectors A = 4i + 3j and B = 1i + 2j, calculate A · B.

A. 10
B. 11
C. 12
D. 13

Solution

A · B = (4)(1) + (3)(2) = 4 + 6 = 10.

Correct Answer: A — 10

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Q. Given vectors A = i + 2j + 3k and B = 2i + 3j + 4k, calculate A · B.

A. 20
B. 22
C. 24
D. 26

Solution

A · B = (1)(2) + (2)(3) + (3)(4) = 2 + 6 + 12 = 20.

Correct Answer: C — 24

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Q. Given vectors A = i + 2j + 3k and B = 4i + 5j + 6k, calculate A · B.

A. 32
B. 30
C. 28
D. 34

Solution

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

Correct Answer: B — 30

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Q. Given vectors A = i + 2j + 3k and B = 4i + 5j + 6k, find A · B. (2019)

A. 32
B. 34
C. 36
D. 38

Solution

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

Correct Answer: B — 34

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Q. Given vectors A = i + 2j and B = 3i + 4j, calculate A · B. (2022)

A. 11
B. 10
C. 12
D. 13

Solution

A · B = (1)(3) + (2)(4) = 3 + 8 = 11.

Correct Answer: A — 11

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Q. If A = 1i + 1j + 1k and B = 1i + 1j + 1k, what is A · B?

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

Solution

A · B = (1)(1) + (1)(1) + (1)(1) = 1 + 1 + 1 = 3.

Correct Answer: C — 3

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

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

Solution

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

Correct Answer: A — 6

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Q. If A = 1i + 1j + 1k and B = 1i + 2j + 3k, what is A · B?

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

Solution

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

Correct Answer: B — 5

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

A. 6
B. 4
C. 2
D. 8

Solution

A · B = (1)(2) + (1)(2) + (1)(2) = 2 + 2 + 2 = 6.

Correct Answer: A — 6

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Q. If A = 1i + 1j and B = 1i - 1j, what is the scalar product A · B?

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

Solution

A · B = (1)(1) + (1)(-1) = 1 - 1 = 0.

Correct Answer: A — 0

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Showing 1 to 30 of 156 (6 Pages)

Vector Algebra

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