Vector & 3D Geometry for Competitive Exam Preparation

Vector & 3D Geometry

3D Lines 3D Planes Scalar Product Vector Algebra Basics Vector Product

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 scalar product of two vectors A and B is 0, what can be said about the vectors?

A. They are parallel
B. They are orthogonal
C. They are equal
D. They are collinear

Solution

If A · B = 0, then the vectors are orthogonal.

Correct Answer: B — They are orthogonal

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Q. If the scalar product of vectors A = (x, y, z) and B = (2, -1, 3) is 10, what is the equation?

A. 2x - y + 3z = 10
B. 2x + y + 3z = 10
C. 2x - y - 3z = 10
D. 2x + y - 3z = 10

Solution

A · B = 2x - y + 3z = 10.

Correct Answer: A — 2x - y + 3z = 10

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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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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 the vectors A = (2, 3) and B = (4, 5) are given, what is the scalar product A · B?

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

Solution

A · B = 2*4 + 3*5 = 8 + 15 = 23.

Correct Answer: C — 20

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Q. If the vectors A = (3, -2, 1) and B = (k, 4, -2) are orthogonal, find the value of k.

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

Solution

A · B = 3k - 8 - 2 = 0; 3k - 10 = 0; k = 10/3.

Correct Answer: A — -1

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Q. If the vectors A = (x, 2, 3) and B = (4, y, 6) are orthogonal, what is the value of y?

A. 2
B. 3
C. 4
D. 5

Solution

A · B = x*4 + 2*y + 3*6 = 0. Thus, 4x + 2y + 18 = 0. If x = 0, then y = -9. If x = 1, y = -10. The only integer solution is y = 3.

Correct Answer: B — 3

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Vector & 3D Geometry

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