Vector Algebra Basics for Competitive Exam Success

Vector Algebra Basics

Q. If the vector a = (2, 2) is scaled by a factor of 3, what is the resulting vector?

A. (6, 6)
B. (3, 3)
C. (2, 2)
D. (1, 1)

Solution

Scaled vector = 3 * a = 3 * (2, 2) = (6, 6)

Correct Answer: A — (6, 6)

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Q. If the vector a = (2, 3) and b = (4, 1), what is the cross product a × b?

A. -10
B. 10
C. 0
D. 5

Solution

Cross product a × b = 2*1 - 3*4 = 2 - 12 = -10.

Correct Answer: A — -10

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Q. If the vector a = (2, 3) and b = (4, 1), what is the resultant vector a + b?

A. (6, 4)
B. (2, 4)
C. (4, 2)
D. (6, 2)

Solution

Resultant vector a + b = (2+4, 3+1) = (6, 4).

Correct Answer: A — (6, 4)

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Q. If the vector A = (2, 3) is multiplied by 2, what is the resulting vector?

A. (4, 6)
B. (2, 3)
C. (1, 1.5)
D. (0, 0)

Solution

Resulting vector = 2 * A = 2 * (2, 3) = (4, 6).

Correct Answer: A — (4, 6)

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Q. If the vector A = (2, 3) is reflected across the line y = x, what is the resulting vector?

A. (3, 2)
B. (2, 3)
C. (0, 0)
D. (1, 1)

Solution

Reflection across y = x gives vector (3, 2).

Correct Answer: A — (3, 2)

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Q. If the vector A = (2, 3) is scaled by a factor of 2, what is the resulting vector?

A. (4, 6)
B. (2, 3)
C. (1, 1.5)
D. (0, 0)

Solution

Scaled vector = 2 * A = 2 * (2, 3) = (4, 6).

Correct Answer: A — (4, 6)

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Q. If the vector a = (2, 3, 4) and b = (1, 0, -1), what is a + b?

A. (3, 3, 3)
B. (1, 3, 3)
C. (2, 3, 3)
D. (2, 3, 5)

Solution

a + b = (2+1, 3+0, 4-1) = (3, 3, 3).

Correct Answer: A — (3, 3, 3)

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Q. If the vector a = (2, 3, 4) and b = (1, 0, -1), what is the scalar triple product a · (b × a)?

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

Solution

The scalar triple product is 0 because a · (b × a) = 0.

Correct Answer: A — 0

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Q. If the vector a = (2, 3, 4) is scaled by a factor of 2, what is the resulting vector?

A. (4, 6, 8)
B. (2, 3, 4)
C. (1, 1.5, 2)
D. (0, 0, 0)

Solution

Scaling the vector a by 2 gives (2*2, 2*3, 2*4) = (4, 6, 8).

Correct Answer: A — (4, 6, 8)

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Q. If the vector a = (3, 4) and b = (1, 2), find the cross product a × b.

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

Solution

In 2D, a × b = a1*b2 - a2*b1 = 3*2 - 4*1 = 6 - 4 = 2.

Correct Answer: A — -2

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Q. If the vector a = (3, 4) is scaled by a factor of 2, what is the new vector?

A. (6, 8)
B. (3, 4)
C. (1.5, 2)
D. (0, 0)

Solution

New vector = 2 * (3, 4) = (6, 8).

Correct Answer: A — (6, 8)

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Q. If the vector a = (3, 4, 0) and b = (0, 0, 5), what is the magnitude of a × b?

A. 15
B. 20
C. 12
D. 10

Solution

Magnitude of a × b = |a||b|sin(90) = |(3, 4, 0)|| (0, 0, 5)| = 5√(3^2 + 4^2) = 15.

Correct Answer: A — 15

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Q. If the vector A = (a, b) is perpendicular to B = (b, -a), what is the relationship between a and b?

A. a = b
B. a = -b
C. a + b = 0
D. a - b = 0

Solution

A·B = ab - ab = 0, hence A and B are perpendicular if a = -b.

Correct Answer: B — a = -b

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Q. If the vectors A = (1, 2) and B = (2, 1) are given, what is the angle between them?

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

Solution

Cosine of angle θ = (A · B) / (|A| |B|) = (1*2 + 2*1) / (√5 * √5) = 4/5, θ = cos⁻¹(4/5).

Correct Answer: B — 45 degrees

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Q. If u = (1, 2) and v = (3, 4), what is the dot product u · v?

A. 10
B. 11
C. 12
D. 7

Solution

Dot product u · v = 1*3 + 2*4 = 3 + 8 = 11.

Correct Answer: A — 10

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Q. If u = (1, 2) and v = (3, 4), what is u + v?

A. (4, 6)
B. (2, 3)
C. (1, 2)
D. (3, 4)

Solution

u + v = (1 + 3, 2 + 4) = (4, 6)

Correct Answer: A — (4, 6)

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Q. If u = (1, 2, 3) and v = (4, 5, 6), what is the dot product u · v?

A. 32
B. 27
C. 26
D. 30

Solution

Dot product u · v = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.

Correct Answer: B — 27

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Q. If u = (2, 3, 1) and v = (1, 0, -1), find the dot product u · v.

A. 5
B. 2
C. 1
D. 0

Solution

u · v = 2*1 + 3*0 + 1*(-1) = 2 + 0 - 1 = 1.

Correct Answer: A — 5

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Q. If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is A + B?

A. (5, 7, 9)
B. (4, 5, 6)
C. (1, 2, 3)
D. (0, 0, 0)

Solution

A + B = (1+4, 2+5, 3+6) = (5, 7, 9).

Correct Answer: A — (5, 7, 9)

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Q. If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is the angle between them?

A. 0 degrees
B. 30 degrees
C. 60 degrees
D. 90 degrees

Solution

Cosine of angle θ = (A . B) / (|A| |B|) = (1*4 + 2*5 + 3*6) / (√14 * √77) = 0, hence θ = 90 degrees.

Correct Answer: D — 90 degrees

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Q. If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is the vector A - B?

A. (-3, -3, -3)
B. (3, 3, 3)
C. (5, 7, 9)
D. (0, 0, 0)

Solution

A - B = (1-4, 2-5, 3-6) = (-3, -3, -3).

Correct Answer: A — (-3, -3, -3)

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Q. If vector A = (2, 2, 2) and vector B = (1, 1, 1), what is the scalar triple product A . (B × A)?

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

Solution

A . (B × A) = 0, since B × A = 0.

Correct Answer: A — 0

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Q. If vector A = (3, -2, 1) and vector B = (1, 4, -3), what is the cross product A × B?

A. (-5, -10, 14)
B. (5, 10, -14)
C. (10, 14, 5)
D. (14, -5, 10)

Solution

A × B = |i  j  k|\n|3 -2  1|\n|1  4 -3| = (-5, -10, 14).

Correct Answer: A — (-5, -10, 14)

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Q. What is the angle between the vectors (1, 0) and (0, 1)?

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

Solution

The angle between (1, 0) and (0, 1) is 90 degrees.

Correct Answer: B — 90 degrees

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Q. What is the angle between the vectors (1, 2, 2) and (2, 1, 2)?

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

Solution

Cosine of angle θ = (u · v) / (|u| |v|). Calculate to find θ = 60 degrees.

Correct Answer: B — 60 degrees

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Q. What is the angle between the vectors a = (1, 2, 2) and b = (2, 0, 2)?

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

Solution

cos(θ) = (a · b) / (|a| |b|). Calculate a · b = 1*2 + 2*0 + 2*2 = 6, |a| = √(1^2 + 2^2 + 2^2) = 3, |b| = √(2^2 + 0^2 + 2^2) = 2√2. Thus, cos(θ) = 6 / (3 * 2√2) = 1/√2, θ = 45 degrees.

Correct Answer: D — 60 degrees

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Q. What is the angle between the vectors u = (1, 0) and v = (0, 1)?

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

Solution

The angle between u and v is 90 degrees since they are perpendicular.

Correct Answer: B — 90 degrees

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Q. What is the angle between vectors A = (1, 0, 0) and B = (0, 1, 0)?

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

Solution

The angle θ = cos⁻¹((A . B) / (|A| |B|)) = cos⁻¹(0) = 90 degrees.

Correct Answer: C — 90 degrees

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Q. What is the cross product of the vectors (1, 0, 0) and (0, 1, 0)?

A. (0, 0, 1)
B. (1, 1, 0)
C. (0, 0, 0)
D. (1, 0, 0)

Solution

Cross product = (1, 0, 0) × (0, 1, 0) = (0, 0, 1).

Correct Answer: A — (0, 0, 1)

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Q. What is the cross product of the vectors (1, 2, 3) and (4, 5, 6)?

A. (-3, 6, -3)
B. (-3, 6, 3)
C. (3, -6, 3)
D. (3, 6, -3)

Solution

Cross product = |i  j  k| |1  2  3| |4  5  6| = (-3, 6, -3).

Correct Answer: A — (-3, 6, -3)

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Showing 31 to 60 of 86 (3 Pages)

Vector Algebra Basics

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