Vector & 3D Geometry for Competitive Exam Preparation

Vector & 3D Geometry

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

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

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

Solution

G · H = 5*1 + (-3)*1 + 2*1 = 5 - 3 + 2 = 4.

Correct Answer: D — 3

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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 scalar triple product of vectors A = (1, 2, 3), B = (4, 5, 6), and C = (7, 8, 9).

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

Solution

Scalar triple product = A · (B × C) = 0, as vectors are coplanar.

Correct Answer: A — 0

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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. Find the value of k if the vectors A = (1, k, 2) and B = (2, 3, 4) are perpendicular.

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

Solution

A · B = 1*2 + k*3 + 2*4 = 0. Thus, 2 + 3k + 8 = 0, so 3k = -10, k = -10/3.

Correct Answer: A — 1

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Q. For the vectors A = (1, 0, 0) and B = (0, 1, 0), what is the scalar product A · B?

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

Solution

A · B = 1*0 + 0*1 + 0*0 = 0.

Correct Answer: A — 0

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Q. For vectors A = (2, 3) and B = (4, 5), find the scalar product A · B.

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

Solution

A · B = 2*4 + 3*5 = 8 + 15 = 23.

Correct Answer: A — 23

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Q. For vectors A = (3, -2, 1) and B = (1, 4, -2), find A · B.

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

Solution

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

Correct Answer: A — -1

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Q. Given A = 3i + 4j and B = 0i + 0j, find A · B.

A. 0
B. 12
C. 7
D. 3

Solution

A · B = (3)(0) + (4)(0) = 0.

Correct Answer: A — 0

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Q. Given vectors A = (2, -1, 3) and B = (4, 0, -2), find A × B.

A. (-1, -10, 4)
B. (1, 10, -4)
C. (10, -1, 4)
D. (10, 1, -4)

Solution

A × B = |i  j  k|\n|2 -1  3|\n|4  0 -2| = (-1, -10, 4)

Correct Answer: A — (-1, -10, 4)

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Q. Given vectors A = (x, y, z) and B = (1, 2, 3), if A · B = 14, what is the value of x + 2y + 3z?

A. 14
B. 10
C. 8
D. 6

Solution

A · B = x*1 + y*2 + z*3 = 14, thus x + 2y + 3z = 14.

Correct Answer: A — 14

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Q. Given vectors P = (4, 0, -3) and Q = (1, 2, 1), find the scalar product P · Q.

A. -1
B. 5
C. 10
D. 2

Solution

P · Q = 4*1 + 0*2 + (-3)*1 = 4 + 0 - 3 = 1.

Correct Answer: B — 5

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Q. Given vectors P = (4, 1, 0) and Q = (0, 2, 3), find the scalar product P · Q.

A. 0
B. 6
C. 8
D. 10

Solution

P · Q = 4*0 + 1*2 + 0*3 = 0 + 2 + 0 = 2.

Correct Answer: B — 6

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Q. Given vectors P = (4, 1, 0) and Q = (1, 2, 3), find the scalar product P · Q.

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

Solution

P · Q = 4*1 + 1*2 + 0*3 = 4 + 2 + 0 = 6.

Correct Answer: B — 11

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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, 0, -1) and B = (0, 1, 1), what is the scalar product A · B?

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

Solution

A · B = 1*0 + 0*1 + (-1)*1 = 0 + 0 - 1 = -1.

Correct Answer: A — 0

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Q. If A = (1, 0, 0) and B = (0, 1, 0), what is the value of A · B?

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

Solution

A · B = 1*0 + 0*1 + 0*0 = 0.

Correct Answer: A — 0

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Q. If A = (1, 0, 0) and B = (0, 1, 0), what is the vector product A × B?

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

Solution

A × B = (0, 0, 1) using the right-hand rule.

Correct Answer: A — (0, 0, 1)

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Q. If A = (1, 1, 1) and B = (1, 1, 1), what is the scalar product A · B?

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

Solution

A · B = 1*1 + 1*1 + 1*1 = 3.

Correct Answer: C — 3

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Q. If A = (1, 1, 1) and B = (2, 2, 2), what is A × B?

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

Solution

A × B = (0, 0, 0) since A and B are parallel.

Correct Answer: A — (0, 0, 0)

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

A. 3
B. 6
C. 9
D. 12

Solution

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

Correct Answer: B — 6

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Q. If A = (1, 1, 1) and B = (x, y, z) such that A · B = 3, what is the value of x + y + z?

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

Solution

A · B = 1*x + 1*y + 1*z = x + y + z = 3.

Correct Answer: C — 3

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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 = (1, 2, 3) and B = (0, 1, 0), what is the direction of the vector product A × B?

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

Solution

A × B = (2, -3, 1) gives direction (2, -3, 1).

Correct Answer: B — (3, 0, -1)

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Q. If A = (1, 2, 3) and B = (4, 5, 6), what is the magnitude of the vector product A × B?

A. 0
B. 1
C. 2√2
D. √14

Solution

Magnitude |A × B| = √(1^2 + 2^2 + 3^2) = √14.

Correct Answer: D — √14

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Showing 31 to 60 of 210 (7 Pages)

Vector & 3D Geometry

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