Vector Algebra Basics for Competitive Exam Success

Vector Algebra Basics

Q. Find the angle between the vectors (1, 0, 0) and (0, 1, 0).

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

Solution

The angle θ = cos⁻¹((u · v) / (|u| |v|)) = cos⁻¹(0) = 90 degrees.

Correct Answer: B — 90 degrees

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Q. Find the cross product of vectors A = (1, 2, 3) and B = (4, 5, 6).

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

Solution

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

Correct Answer: A — (-3, 6, -3)

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Q. Find the magnitude of the vector (3, 4).

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

Solution

Magnitude = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.

Correct Answer: A — 5

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Q. Find the magnitude of the vector v = (3, -4, 12).

A. 13
B. 14
C. 15
D. 12

Solution

Magnitude |v| = √(3^2 + (-4)^2 + 12^2) = √(9 + 16 + 144) = √169 = 13.

Correct Answer: B — 14

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Q. Find the scalar projection of vector A = (3, 4) onto vector B = (1, 0).

A. 3
B. 4
C. 1
D. 0

Solution

Scalar projection = (A · B) / |B| = (3*1 + 4*0) / 1 = 3.

Correct Answer: A — 3

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Q. Find the unit vector in the direction of the vector (3, 4).

A. (0.6, 0.8)
B. (0.8, 0.6)
C. (1, 1)
D. (0.5, 0.5)

Solution

Magnitude = √(3^2 + 4^2) = 5. Unit vector = (3/5, 4/5) = (0.6, 0.8).

Correct Answer: A — (0.6, 0.8)

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Q. Find the unit vector in the direction of the vector (3, 4, 0).

A. (0.6, 0.8, 0)
B. (0.3, 0.4, 0)
C. (1, 1, 0)
D. (0, 0, 1)

Solution

Magnitude = √(3^2 + 4^2) = 5. Unit vector = (3/5, 4/5, 0) = (0.6, 0.8, 0).

Correct Answer: A — (0.6, 0.8, 0)

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Q. Find the unit vector in the direction of the vector (4, 3).

A. (4/5, 3/5)
B. (3/5, 4/5)
C. (1, 0)
D. (0, 1)

Solution

Unit vector = (4, 3) / √(4^2 + 3^2) = (4, 3) / 5 = (4/5, 3/5).

Correct Answer: A — (4/5, 3/5)

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Q. Find the unit vector in the direction of the vector (6, 8).

A. (0.6, 0.8)
B. (0.8, 0.6)
C. (1, 1)
D. (0.5, 0.5)

Solution

Magnitude = √(6^2 + 8^2) = √(36 + 64) = √100 = 10. Unit vector = (6/10, 8/10) = (0.6, 0.8).

Correct Answer: A — (0.6, 0.8)

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Q. Find the unit vector in the direction of the vector v = (4, -3).

A. (4/5, -3/5)
B. (3/5, 4/5)
C. (4/3, -3/4)
D. (3/4, 4/3)

Solution

Magnitude |v| = √(4^2 + (-3)^2) = √(16 + 9) = 5. Unit vector = (4/5, -3/5).

Correct Answer: A — (4/5, -3/5)

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Q. If A = (1, 0) and B = (0, 1), what is the angle between them?

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

Solution

Angle = cos⁻¹((A·B) / (|A||B|)) = cos⁻¹(0) = 90 degrees

Correct Answer: B — 90 degrees

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

A. 10
B. 11
C. 12
D. 8

Solution

Dot product A · B = 1*3 + 2*4 = 3 + 8 = 11.

Correct Answer: A — 10

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Q. If A = (1, 2) and B = (3, 4), what is the midpoint M of AB?

A. (2, 3)
B. (1, 2)
C. (3, 4)
D. (4, 5)

Solution

Midpoint M = ((1+3)/2, (2+4)/2) = (2, 3).

Correct Answer: A — (2, 3)

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

A. (2, 2)
B. (2, 3)
C. (4, 5)
D. (6, 8)

Solution

AB = B - A = (4 - 2, 5 - 3) = (2, 2)

Correct Answer: A — (2, 2)

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Q. If A = (2, 3) and B = (4, 7), find the vector AB.

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

Solution

Vector AB = B - A = (4 - 2, 7 - 3) = (2, 4).

Correct Answer: A — (2, 4)

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Q. If A(1, 2, 3) and B(4, 5, 6) are two points in space, what is the vector AB?

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

Solution

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

Correct Answer: A — (3, 3, 3)

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Q. If A(1, 2, 3) and B(4, 5, 6) are two points, what is the vector AB?

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

Solution

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

Correct Answer: A — (3, 3, 3)

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Q. If A(2, 3, 4) and B(1, 0, -1) are two points in space, find the vector AB.

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

Solution

AB = B - A = (1 - 2, 0 - 3, -1 - 4) = (-1, -3, -5) = (1, 3, 5) in the opposite direction.

Correct Answer: A — (1, 3, 5)

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Q. If the position vector of a point is (5, 12), what is its distance from the origin?

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

Solution

Distance = √(5^2 + 12^2) = √(25 + 144) = √169 = 13

Correct Answer: A — 13

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Q. If the position vector of a point is given by r = (2t, 3t, 4t), what is the velocity vector?

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

Solution

Velocity vector = dr/dt = (2, 3, 4)

Correct Answer: A — (2, 3, 4)

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Q. If the position vector of a point P is (2, 3, 4), what is the distance from the origin to point P?

A. 5
B. 6
C. 7
D. 8

Solution

Distance = √(2^2 + 3^2 + 4^2) = √(4 + 9 + 16) = √29 ≈ 5.385.

Correct Answer: B — 6

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Q. If the position vector of a point P is (x, y, z) and the vector a = (1, 2, 3), what is the projection of P onto a?

A. (1, 2, 3)
B. (2, 4, 6)
C. (0, 0, 0)
D. (x, y, z)

Solution

Projection of P onto a = ((P · a) / |a|^2) * a.

Correct Answer: D — (x, y, z)

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Q. If the position vector of a point P is given by r = (2t, 3t, 4t), find the coordinates of P when t = 1.

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

Solution

Substituting t = 1, r = (2*1, 3*1, 4*1) = (2, 3, 4).

Correct Answer: A — (2, 3, 4)

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Q. If the position vector of point P is (3, -2) and Q is (1, 4), what is the vector PQ?

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

Solution

Vector PQ = Q - P = (1 - 3, 4 - (-2)) = (-2, 6).

Correct Answer: A — (-2, 6)

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Q. If the position vector of point P is (3, 4) and Q is (1, 2), what is the vector PQ?

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

Solution

Vector PQ = Q - P = (1 - 3, 2 - 4) = (-2, -2).

Correct Answer: A — (2, 2)

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

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

Solution

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

Correct Answer: C — 45 degrees

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Q. If the vector a = (1, 2) and b = (3, 4), find the angle between them using the dot product.

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

Solution

cos(θ) = (a · b) / (|a| |b|). a · b = 1*3 + 2*4 = 11, |a| = √(1^2 + 2^2) = √5, |b| = √(3^2 + 4^2) = 5. Thus, cos(θ) = 11 / (√5 * 5) = 11 / (5√5), θ = 60 degrees.

Correct Answer: D — 60 degrees

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

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

Solution

a + b = (2 + 1, -1 + 3) = (3, 2)

Correct Answer: A — (3, 2)

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

A. 5
B. 7
C. 1
D. 0

Solution

Cross product in 2D = a1*b2 - a2*b1 = 2*3 - (-1)*1 = 6 + 1 = 7

Correct Answer: A — 5

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Q. If the vector a = (2, 2) and b = (2, -2), what is the angle between them?

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

Solution

Angle = cos⁻¹((a·b) / (|a||b|)) = cos⁻¹(0) = 90 degrees

Correct Answer: A — 90 degrees

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Vector Algebra Basics

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