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

Learn More →

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

Learn More →

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

Learn More →

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

Learn More →

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

Learn More →

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

Learn More →

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)

Learn More →

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

Learn More →

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

Learn More →

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

Learn More →

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°

Learn More →

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°

Learn More →

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°

Learn More →

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°

Learn More →

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

Learn More →

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)

Learn More →

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

Learn More →

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

Learn More →

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

Learn More →

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

Learn More →

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

Learn More →

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

Learn More →

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

Learn More →

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

Learn More →

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

Learn More →

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

Learn More →

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

Learn More →

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)

Learn More →

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)

Learn More →

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)

Learn More →

Showing 1 to 30 of 170 (6 Pages)