Vector & 3D Geometry for Competitive Exam Preparation

Vector & 3D Geometry

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

Q. What is 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. What is the magnitude of the vector C = (6, 8, 10)?

A. 10
B. 12
C. 14
D. 16

Solution

Magnitude |C| = √(6^2 + 8^2 + 10^2) = √(36 + 64 + 100) = √200 = 10√2.

Correct Answer: C — 14

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Q. What is the magnitude of the vector v = (3, -4)?

A. 5
B. 7
C. 4
D. 3

Solution

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

Correct Answer: A — 5

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Q. What is the projection of vector a = (3, 4) onto vector b = (1, 0)?

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

Solution

Projection = (a · b / |b|^2) * b = (3*1 + 4*0) / 1^2 * (1, 0) = 3 * (1, 0) = (3, 0).

Correct Answer: A — 3

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

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

Solution

Projection = (A . B / |B|^2) * B = (3*1 + 4*2) / (1^2 + 2^2) * (1, 2) = 11/5 * (1, 2) = (2.2, 4.4).

Correct Answer: B — 3

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

A. (1, 7)
B. (3, 1)
C. (1, 1)
D. (2, 4)

Solution

Resultant = (2 + (-1), 3 + 4) = (1, 7).

Correct Answer: A — (1, 7)

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Q. What is the resultant of vectors (1, 2) and (-1, -2)?

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

Solution

Resultant = (1 - 1, 2 - 2) = (0, 0)

Correct Answer: A — (0, 0)

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Q. What is the scalar product of A = (3, 4, 0) and B = (0, 0, 5)?

A. 0
B. 15
C. 20
D. 25

Solution

A · B = 3*0 + 4*0 + 0*5 = 0.

Correct Answer: A — 0

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Q. What is the scalar product of the unit vectors i and j?

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

Solution

i · j = 0, since they are orthogonal.

Correct Answer: B — 0

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Q. What is the scalar product of the vectors (3, 4) and (4, 3)?

A. 12
B. 25
C. 24
D. 21

Solution

The scalar product is 3*4 + 4*3 = 12 + 12 = 24.

Correct Answer: B — 25

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

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

Solution

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

Correct Answer: D — 6

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Q. What is the scalar product of the vectors (5, -3) and (-2, 4)?

A. -6
B. 10
C. 22
D. 26

Solution

Scalar product = 5*(-2) + (-3)*4 = -10 - 12 = -22.

Correct Answer: A — -6

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

A. 30
B. 35
C. 40
D. 45

Solution

Scalar product = 5*1 + 5*2 + 5*3 = 5 + 10 + 15 = 30.

Correct Answer: A — 30

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Q. What is the scalar product of the vectors A = (0, 1, 0) and B = (1, 0, 1)?

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

Solution

A · B = 0*1 + 1*0 + 0*1 = 0.

Correct Answer: A — 0

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Q. What is the scalar product of the vectors A = (1, 1, 1) and B = (1, 1, 1)?

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

Solution

A · B = 1*1 + 1*1 + 1*1 = 1 + 1 + 1 = 3.

Correct Answer: C — 3

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

A. -10
B. 10
C. 8
D. 0

Solution

A · B = 2*0 + (-1)*4 + 3*(-2) = 0 - 4 - 6 = -10.

Correct Answer: A — -10

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Q. What is the scalar product of the vectors A = (4, 0, -3) and B = (0, 5, 2)?

A. -6
B. 0
C. 8
D. 20

Solution

A · B = 4*0 + 0*5 + (-3)*2 = 0 - 6 = -6.

Correct Answer: B — 0

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Q. What is the scalar product of the vectors K = (0, 1, 0) and L = (1, 0, 1)?

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

Solution

K · L = 0*1 + 1*0 + 0*1 = 0 + 0 + 0 = 0.

Correct Answer: A — 0

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Q. What is the scalar projection of vector (3, 4) onto (1, 0)?

A. 3
B. 4
C. 5
D. 0

Solution

Scalar projection = (3*1 + 4*0) / √(1^2) = 3

Correct Answer: A — 3

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

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

Solution

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

Correct Answer: A — 3

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Q. What is the scalar triple product of the vectors (1, 2, 3), (4, 5, 6), and (7, 8, 9)?

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

Solution

Scalar triple product = (1, 2, 3) · ((4, 5, 6) × (7, 8, 9)) = 0.

Correct Answer: A — 0

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Q. What is the scalar triple product of vectors A = (1, 0, 0), B = (0, 1, 0), C = (0, 0, 1)?

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

Solution

Scalar triple product = A · (B × C) = 1.

Correct Answer: A — 1

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Q. What is the scalar triple product of vectors a = (1, 2, 3), b = (4, 5, 6), c = (7, 8, 9)?

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

Solution

Scalar triple product = a · (b × c). Since b and c are linearly dependent, b × c = 0, hence the scalar triple product is 0.

Correct Answer: A — 0

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Q. What is the unit vector in the direction of (3, 4)?

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

Solution

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

Correct Answer: A — (3/5, 4/5)

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Q. What is the unit vector in the direction of (3, 4, 0)?

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

Solution

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

Correct Answer: A — (0.6, 0.8, 0)

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Q. What is the unit vector in the direction of (4, 3)?

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

Solution

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

Correct Answer: A — (4/5, 3/5)

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Q. What is 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 = (4/5, 3/5) since magnitude = 5.

Correct Answer: A — (4/5, 3/5)

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Q. What is the unit vector in the direction of v = (3, 4)?

A. (0.6, 0.8)
B. (1, 1)
C. (3, 4)
D. (0, 0)

Solution

Unit vector = v / |v| = (3, 4) / 5 = (0.6, 0.8)

Correct Answer: A — (0.6, 0.8)

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Q. What is the unit vector in the direction of vector A = (3, 4)?

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

Solution

Unit vector = A / |A| = (3, 4) / 5 = (0.6, 0.8).

Correct Answer: A — (0.6, 0.8)

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Vector & 3D Geometry

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