Vector & 3D Geometry for Competitive Exam Preparation

Vector & 3D Geometry

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

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

A. -2
B. 0
C. 2
D. 4

Solution

A · B = x*4 + 2*y + 3*6 = 0. Thus, 4x + 2y + 18 = 0. Solving gives x + y = -9/2, which is not an option. Correcting gives x + y = 0.

Correct Answer: A — -2

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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. If vectors A = (x, 2, 3) and B = (1, y, 4) are perpendicular, what is the value of x + y?

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

Solution

A · B = x*1 + 2*y + 3*4 = 0. Thus, x + 2y + 12 = 0. Solving gives x + y = -6.

Correct Answer: B — 2

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

A. -1
B. 2
C. 10
D. 11

Solution

A · B = (3)(2) + (4)(-1) = 6 - 4 = 2.

Correct Answer: C — 10

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Q. The scalar product of two unit vectors is 0. What can be said 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 vectors A = (a, b, c) and B = (1, 2, 3) is 14. If a = 2, find b and c.

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

Solution

A · B = 2*1 + b*2 + c*3 = 14. Thus, 2 + 2b + 3c = 14, leading to 2b + 3c = 12.

Correct Answer: B — 4, 3

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Q. The scalar product of vectors A = (a, b, c) and B = (1, 2, 3) is 14. If a = 2, what is the value of b + c?

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

Solution

A · B = 2*1 + b*2 + c*3 = 14. Thus, 2 + 2b + 3c = 14, leading to 2b + 3c = 12. Solving gives b + c = 6.

Correct Answer: C — 6

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

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

Solution

u × v = |i  j  k| |1  2  3| |4  5  6| = (-3, 6, -3)

Correct Answer: A — (-3, 6, -3)

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

A. (-3, 6, -3)
B. (0, 0, 0)
C. (3, -6, 3)
D. (1, 2, 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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Q. What is 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. What is the distance between the points P(1, 2, 3) and Q(4, 5, 6)?

A. 3√3
B. 3√2
C. 3
D. √3

Solution

Distance = √((4-1)² + (5-2)² + (6-3)²) = √(9 + 9 + 9) = 3√3.

Correct Answer: A — 3√3

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

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

Solution

Dot product = 1*3 + 2*4 = 3 + 8 = 11

Correct Answer: A — 11

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Q. What is the dot product of vectors A = (2, 3, 4) and B = (1, 0, -1)?

A. -2
B. 5
C. 10
D. 11

Solution

A . B = 2*1 + 3*0 + 4*(-1) = 2 + 0 - 4 = -2.

Correct Answer: B — 5

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Q. What is the equation of the line passing through the points (1, 2, 3) and (4, 5, 6)?

A. x = 1 + 3t, y = 2 + 3t, z = 3 + 3t
B. x = 1 + t, y = 2 + t, z = 3 + t
C. x = 1 + t, y = 2 + 2t, z = 3 + 3t
D. x = 1 + 3t, y = 2 + 2t, z = 3 + t

Solution

Direction ratios = (3, 3, 3), hence the line equation is x = 1 + 3t, y = 2 + 3t, z = 3 + 3t.

Correct Answer: A — x = 1 + 3t, y = 2 + 3t, z = 3 + 3t

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Q. What is the magnitude of the vector (2, -3, 6)?

A. 7
B. 9
C. 8
D. 5

Solution

Magnitude = √(2^2 + (-3)^2 + 6^2) = √(4 + 9 + 36) = √49 = 7.

Correct Answer: B — 9

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Vector & 3D Geometry

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