Vector Algebra for Competitive Exam Success

Vector Algebra

Scalar Product Vector Product Vectors in 2D & 3D

Q. If the position vector of point P is given by r = 2i + 3j + 4k, what is the x-coordinate of point P? (2020)

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

Solution

The x-coordinate of point P is the coefficient of i in the position vector, which is 2.

Correct Answer: A — 2

Learn More →

Q. If the position vector of point P is given by r = 2i + 3j + 4k, what is the x-component of r?

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

Solution

The x-component of r is 2.

Correct Answer: A — 2

Learn More →

Q. If the position vector of point P is given by r = 3i + 4j, what is the distance of point P from the origin?

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

Solution

Distance = |r| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.

Correct Answer: A — 5

Learn More →

Q. If the scalar product of two vectors A and B is 0, what can be inferred about the vectors?

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

Solution

If A · B = 0, then A and B are orthogonal (perpendicular) to each other.

Correct Answer: C — They are orthogonal

Learn More →

Q. If the scalar product of two vectors A and B is 15 and the magnitudes are |A| = 5 and |B| = 3, find the angle between them.

A. 60°
B. 45°
C. 30°
D. 90°

Solution

A · B = |A||B|cos(θ) => 15 = 5*3*cos(θ) => cos(θ) = 1, θ = 0°.

Correct Answer: B — 45°

Learn More →

Q. If the scalar product of two vectors A and B is equal to the product of their magnitudes, what can be inferred?

A. They are perpendicular
B. They are parallel
C. They are equal
D. They are opposite

Solution

If A · B = |A||B|, then the angle between them is 0°, meaning they are parallel.

Correct Answer: B — They are parallel

Learn More →

Q. If the scalar product of vectors A and B is equal to the product of their magnitudes, what can be said about the angle between them?

A. 0°
B. 90°
C. 180°
D. 45°

Solution

If A · B = |A||B|, then cos(θ) = 1, which means θ = 0°.

Correct Answer: A — 0°

Learn More →

Q. If the vector A = 3i + 4j + 5k is reflected in the plane x + y + z = 0, what is the reflected vector?

A. -3i - 4j - 5k
B. 3i + 4j + 5k
C. 0
D. None of the above

Solution

The reflection of A in the plane x + y + z = 0 is -A = -3i - 4j - 5k.

Correct Answer: A — -3i - 4j - 5k

Learn More →

Q. If the vector A = 5i + 12j is scaled by a factor of 2, what is the new vector?

A. 10i + 24j
B. 5i + 12j
C. 2i + 3j
D. 7i + 14j

Solution

Scaling A by 2 gives 2A = 2(5i + 12j) = 10i + 24j.

Correct Answer: A — 10i + 24j

Learn More →

Q. If the vectors A = 1i + 2j and B = 2i + 1j, find A · B. (2023)

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

Solution

A · B = (1)(2) + (2)(1) = 2 + 2 = 4

Correct Answer: B — 5

Learn More →

Q. If the vectors A = 2i + 2j and B = 2i - 2j, find A · B.

A. 0
B. 4
C. 8
D. 2

Solution

A · B = (2)(2) + (2)(-2) = 4 - 4 = 0.

Correct Answer: B — 4

Learn More →

Q. If the vectors A = 2i + 3j and B = 3i + 4j are perpendicular, what is the value of A · B?

A. 0
B. 6
C. 12
D. 9

Solution

Since A and B are perpendicular, A · B = 0.

Correct Answer: A — 0

Learn More →

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

A. 25
B. 30
C. 32
D. 28

Solution

A · B = (3)(4) + (4)(3) = 12 + 12 = 24.

Correct Answer: A — 25

Learn More →

Q. If the vectors A = 5i + 5j and B = 5i - 5j, what is the scalar product A · B?

A. 50
B. 0
C. 25
D. 10

Solution

A · B = (5)(5) + (5)(-5) = 25 - 25 = 0.

Correct Answer: B — 0

Learn More →

Q. If the vectors A = 6i + 8j and B = 2i + 3j, what is A · B? (2020)

A. 42
B. 48
C. 36
D. 30

Solution

A · B = (6)(2) + (8)(3) = 12 + 24 = 36.

Correct Answer: A — 42

Learn More →

Q. If the vectors A = 6i + 8j and B = 3i + 4j are perpendicular, what is the value of A · B? (2021)

A. 0
B. 18
C. 30
D. 42

Solution

A · B = (6)(3) + (8)(4) = 18 + 32 = 50, but since they are perpendicular, A · B = 0.

Correct Answer: A — 0

Learn More →

Q. If the vectors A and B are such that A · B = |A| |B|, what is the angle between them?

A. 0°
B. 90°
C. 180°
D. None of the above

Solution

If A · B = |A| |B|, then cos(θ) = 1, which means θ = 0°.

Correct Answer: A — 0°

Learn More →

Q. If vector A = 2i + 3j + k and vector B = i - 2j + 4k, what is the cross product A × B?

A. -10i + 5j + 7k
B. 10i - 5j - 7k
C. 10i + 5j + 7k
D. -10i - 5j + 7k

Solution

A × B = |i  j  k|\n|2  3  1|\n|1 -2  4| = (3*4 - 1*(-2))i - (2*4 - 1*1)j + (2*(-2) - 3*1)k = 10i - 5j - 7k.

Correct Answer: A — -10i + 5j + 7k

Learn More →

Q. If vector A = 2i + 3j and vector B = -i + 4j, what is the resultant vector R = A + B?

A. i + 7j
B. i + j
C. 3i + 7j
D. 3i + 4j

Solution

R = A + B = (2 - 1)i + (3 + 4)j = 1i + 7j.

Correct Answer: A — i + 7j

Learn More →

Q. If vector A = 2i + 3j and vector B = 3i + 4j, what is the angle between them? (2023)

A. 0 degrees
B. 90 degrees
C. 45 degrees
D. 60 degrees

Solution

cos(θ) = (A · B) / (|A| |B|) = (2*3 + 3*4) / (√(2^2 + 3^2) * √(3^2 + 4^2)) = 0.6, θ = 53.13 degrees.

Correct Answer: D — 60 degrees

Learn More →

Q. If vector A = 2i + 3j and vector B = 4i + 5j, what is the angle between A and B? (2023)

A. 0 degrees
B. 90 degrees
C. 45 degrees
D. 60 degrees

Solution

cos(θ) = (A · B) / (|A||B|) = (8 + 15) / (√(13) * √(41)). θ = 60 degrees.

Correct Answer: D — 60 degrees

Learn More →

Q. If vector A = 2i + 3j and vector B = 4i + 5j, what is the angle between them?

A. 0 degrees
B. 90 degrees
C. 45 degrees
D. 60 degrees

Solution

cos(θ) = (A · B) / (|A||B|) = (2*4 + 3*5) / (√(2^2 + 3^2) * √(4^2 + 5^2)) = 0.5, θ = 60 degrees.

Correct Answer: D — 60 degrees

Learn More →

Q. If vector A = 2i + 3j and vector B = 4i + 6j, are the vectors A and B parallel?

A. Yes
B. No
C. Cannot be determined
D. Only if scaled

Solution

Vectors A and B are parallel because B = 2A.

Correct Answer: A — Yes

Learn More →

Q. If vector A = 2i + 3j and vector B = 4i + k, what is the angle between A and B?

A. 90 degrees
B. 60 degrees
C. 45 degrees
D. 30 degrees

Solution

cos(θ) = (A · B) / (|A| |B|). A · B = 8 + 0 + 3(0) = 8; |A| = √(2^2 + 3^2) = √13; |B| = √(4^2 + 1^2) = √17. θ = cos^(-1)(8/(√13 * √17)).

Correct Answer: B — 60 degrees

Learn More →

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

A. 10
B. 5
C. 7
D. 8

Solution

A · B = (3)(2) + (4)(-1) = 6 - 4 = 2.

Correct Answer: A — 10

Learn More →

Q. If vector A = 3i + 4j and vector B = 4i + 3j, what is the cross product A × B?

A. -1k
B. 1k
C. 0
D. 7k

Solution

A × B = |i  j  k| |3  4  0| |4  3  0| = (0 - 0)i - (0 - 0)j + (9 - 12)k = -3k.

Correct Answer: B — 1k

Learn More →

Q. If vector A = 4i + 3j and vector B = -3i + 4j, what is the angle between them?

A. 90 degrees
B. 45 degrees
C. 60 degrees
D. 30 degrees

Solution

cos(θ) = (A · B) / (|A||B|) = (4*-3 + 3*4) / (5*5) = 0, θ = 90 degrees.

Correct Answer: A — 90 degrees

Learn More →

Q. If vector A = 4i + 3j and vector B = -i + 2j, what is the resultant vector A + B? (2019)

A. 3i + 5j
B. 5i + j
C. 3i + j
D. 5i + 5j

Solution

A + B = (4 - 1)i + (3 + 2)j = 3i + 5j.

Correct Answer: A — 3i + 5j

Learn More →

Q. If vector A = 4i + 3j and vector B = -i + 2j, what is the resultant vector R = A + B? (2019)

A. 3i + 5j
B. 5i + j
C. 3i + j
D. 5i + 5j

Solution

R = A + B = (4 - 1)i + (3 + 2)j = 3i + 5j.

Correct Answer: A — 3i + 5j

Learn More →

Q. If vector A = 4i + 3j and vector B = 3i + 4j, what is the angle between A and B?

A. 45 degrees
B. 60 degrees
C. 90 degrees
D. 135 degrees

Solution

cos(θ) = (A · B) / (|A||B|) = (12 + 12) / (5 * 5) = 24/25, θ = cos^(-1)(24/25) ≈ 60 degrees.

Correct Answer: B — 60 degrees

Learn More →

Showing 91 to 120 of 156 (6 Pages)

Vector Algebra

On a scale of 0–10, how likely are you to recommend The Soulshift Academy?