Vector & 3D Geometry for Competitive Exam Preparation

Vector & 3D Geometry

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

Q. Calculate the scalar product of A = (1, 1, 1) and B = (2, 2, 2).

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

Solution

A · B = 1*2 + 1*2 + 1*2 = 2 + 2 + 2 = 6.

Correct Answer: D — 6

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Q. Calculate the scalar product of the vectors (1, 0, 0) and (0, 1, 0).

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

Solution

Scalar product = 1*0 + 0*1 + 0*0 = 0.

Correct Answer: A — 0

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Q. Calculate the scalar product of the vectors (1, 2, 3) and (4, 5, 6).

A. 32
B. 33
C. 34
D. 35

Solution

Scalar product = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.

Correct Answer: A — 32

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Q. Calculate the scalar product of the vectors (2, 3, 4) and (4, 3, 2).

A. 28
B. 29
C. 30
D. 31

Solution

Scalar product = 2*4 + 3*3 + 4*2 = 8 + 9 + 8 = 25.

Correct Answer: A — 28

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Q. Calculate the scalar product of the vectors (3, 0, -3) and (1, 2, 1).

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

Solution

Scalar product = 3*1 + 0*2 + (-3)*1 = 3 + 0 - 3 = 0.

Correct Answer: A — 0

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

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

Solution

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

Correct Answer: B — 30

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Q. Calculate the scalar product of the vectors A = (4, -1, 2) and B = (2, 3, 1).

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

Solution

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

Correct Answer: A — 10

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Q. Calculate the scalar product of the vectors K = (0, 1, 2) and L = (3, 4, 5).

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

Solution

K · L = 0*3 + 1*4 + 2*5 = 0 + 4 + 10 = 14.

Correct Answer: A — 10

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Q. Calculate the vector product of A = (3, 2, 1) and B = (1, 0, 2).

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

Solution

A × B = |i  j  k|\n|3  2  1|\n|1  0  2| = (4, 5, -2)

Correct Answer: A — (4, 5, -2)

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Q. Determine the scalar product of the vectors (0, 1, 2) and (3, 4, 5).

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

Solution

Scalar product = 0*3 + 1*4 + 2*5 = 0 + 4 + 10 = 14.

Correct Answer: B — 11

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Q. Determine the scalar product of the vectors A = (1, 1, 1) and B = (2, 2, 2).

A. 3
B. 4
C. 6
D. 8

Solution

A · B = 1*2 + 1*2 + 1*2 = 2 + 2 + 2 = 6.

Correct Answer: C — 6

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Q. Determine the scalar product of the vectors A = (2, 2, 2) and B = (3, 3, 3).

A. 12
B. 18
C. 6
D. 9

Solution

A · B = 2*3 + 2*3 + 2*3 = 6 + 6 + 6 = 18.

Correct Answer: A — 12

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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 angle between the vectors A = (1, 2, 2) and B = (2, 0, 2).

A. 0°
B. 45°
C. 60°
D. 90°

Solution

cos(θ) = (A · B) / (|A| |B|). 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. cos(θ) = 6 / (3 * 2√2) = 1/√2, θ = 45°.

Correct Answer: C — 60°

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Q. Find the angle between the vectors A = (1, 2, 2) and B = (2, 1, 1).

A. 60°
B. 45°
C. 30°
D. 90°

Solution

cos(θ) = (A · B) / (|A| |B|). A · B = 1*2 + 2*1 + 2*1 = 6; |A| = √(1^2 + 2^2 + 2^2) = 3; |B| = √(2^2 + 1^2 + 1^2) = √6. Thus, cos(θ) = 6 / (3√6) = 1/√6, θ = 45°.

Correct Answer: B — 45°

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Q. Find the angle between the vectors A = (3, -2, 1) and B = (1, 1, 1) if A · B = |A||B|cos(θ).

A. 60°
B. 45°
C. 90°
D. 30°

Solution

A · B = 3*1 + (-2)*1 + 1*1 = 3 - 2 + 1 = 2. |A| = √(3^2 + (-2)^2 + 1^2) = √14, |B| = √3. cos(θ) = 2/(√14 * √3). θ = 60°.

Correct Answer: A — 60°

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Q. Find the angle between the vectors A = (3, -2, 1) and B = (1, 1, 1).

A. 60°
B. 45°
C. 90°
D. 30°

Solution

cos(θ) = (A · B) / (|A| |B|). A · B = 3*1 + (-2)*1 + 1*1 = 2. |A| = √(3^2 + (-2)^2 + 1^2) = √14, |B| = √3. θ = cos^(-1)(2/(√14 * √3)).

Correct Answer: A — 60°

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Q. Find the area of the triangle formed by the points A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9) using the vector product.

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

Solution

Area = 0.5 * |AB × AC| = 0, as points are collinear.

Correct Answer: A — 0

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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 projection of vector A = (2, 3) onto vector B = (1, 1).

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

Solution

Projection of A onto B = (A · B) / |B|^2 * B; A · B = 2*1 + 3*1 = 5; |B|^2 = 1^2 + 1^2 = 2; Projection = (5/2)(1, 1) = (2.5, 2.5).

Correct Answer: A — 1

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Q. Find the projection of vector A = (3, 4) onto vector B = (1, 2).

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

Solution

Projection of A onto B = (A · B) / |B|^2 * B. A · B = 3*1 + 4*2 = 11, |B|^2 = 1^2 + 2^2 = 5. Thus, projection = (11/5) * (1, 2) = (11/5, 22/5).

Correct Answer: B — 2

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

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

Solution

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

Correct Answer: B — 30

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Q. Find the scalar product of the vectors (3, -2, 5) and (1, 4, -1).

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

Solution

Scalar product = 3*1 + (-2)*4 + 5*(-1) = 3 - 8 - 5 = -10.

Correct Answer: A — -1

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Q. Find the scalar product of the vectors (4, 5) and (1, 2).

A. 14
B. 13
C. 12
D. 11

Solution

Scalar product = 4*1 + 5*2 = 4 + 10 = 14.

Correct Answer: A — 14

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Q. Find the scalar product of the vectors (7, 8, 9) and (0, 1, 2).

A. 26
B. 27
C. 28
D. 29

Solution

Scalar product = 7*0 + 8*1 + 9*2 = 0 + 8 + 18 = 26.

Correct Answer: A — 26

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Q. Find the scalar product of the vectors A = (2, 3) and B = (4, -1).

A. -1
B. 5
C. 10
D. 11

Solution

A · B = 2*4 + 3*(-1) = 8 - 3 = 5.

Correct Answer: C — 10

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Q. Find the scalar product of the vectors A = 5i + 12j and B = 3i - 4j.

A. -33
B. 33
C. 39
D. 45

Solution

A · B = (5)(3) + (12)(-4) = 15 - 48 = -33.

Correct Answer: A — -33

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Q. Find the scalar product of the vectors G = (2, -3, 1) and H = (4, 0, -2).

A. -2
B. 0
C. 2
D. 8

Solution

G · H = 2*4 + (-3)*0 + 1*(-2) = 8 + 0 - 2 = 6.

Correct Answer: A — -2

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Vector & 3D Geometry

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