Vector Algebra for Competitive Exam Success

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Vector Algebra MCQ & Objective Questions

Vector Algebra is a crucial topic in mathematics that plays a significant role in various school and competitive exams. Mastering this subject not only enhances your understanding of mathematical concepts but also boosts your confidence in solving objective questions. Practicing MCQs and important questions in Vector Algebra can greatly improve your exam preparation and help you score better.

What You Will Practise Here

Understanding vector addition and subtraction
Scalar and vector products
Applications of vectors in geometry
Key formulas related to vector magnitudes and directions
Representation of vectors in different coordinate systems
Concept of unit vectors and their significance
Solving problems involving vector equations

Exam Relevance

Vector Algebra is frequently tested in various examinations, including CBSE, State Boards, NEET, and JEE. Students can expect questions that involve calculations, conceptual understanding, and application of vector principles. Common question patterns include solving for resultant vectors, determining angles between vectors, and applying vector operations in real-world scenarios.

Common Mistakes Students Make

Confusing scalar and vector quantities
Misapplying vector addition and subtraction rules
Neglecting the importance of direction in vector problems
Overlooking the significance of unit vectors
Failing to visualize vectors geometrically

FAQs

Question: What are some important Vector Algebra MCQ questions I should focus on?
Answer: Focus on questions related to vector addition, scalar and vector products, and applications in geometry.

Question: How can I improve my understanding of Vector Algebra for exams?
Answer: Regular practice of objective questions and solving previous years' exam papers can significantly enhance your understanding.

Start solving practice MCQs today to test your understanding of Vector Algebra and prepare effectively for your exams. The more you practice, the more confident you will become in tackling this essential topic!

Scalar Product Vector Product Vectors in 2D & 3D

Q. Find the angle between the vectors A = 2i + 2j and B = 2i - 2j. (2022)

A. 0 degrees
B. 45 degrees
C. 90 degrees
D. 180 degrees

Solution

cos(θ) = (A · B) / (|A||B|). A · B = 0, hence θ = 90 degrees.

Correct Answer: C — 90 degrees

Q. Find the angle between the vectors A = i + j and B = 2i + 2j.

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

Solution

cos(θ) = (A · B) / (|A| |B|). A · B = 1*2 + 1*2 = 4; |A| = √2, |B| = 2√2. Thus, cos(θ) = 4 / (√2 * 2√2) = 1, θ = 0 degrees.

Correct Answer: A — 0 degrees

Q. Find the angle between the vectors A = i + j and B = i - j.

A. 0 degrees
B. 45 degrees
C. 90 degrees
D. 135 degrees

Solution

cos(θ) = (A · B) / (|A||B|) = (1 - 1) / (√2 * √2) = 0, θ = 90 degrees.

Correct Answer: C — 90 degrees

Q. Find the angle between the vectors A = i + j and B = j - i. (2022)

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

Solution

cos(θ) = (A · B) / (|A| |B|). A · B = 0, hence θ = 90 degrees.

Correct Answer: A — 90 degrees

Q. Find the angle θ between the vectors A = i + 2j and B = 2i + 3j if A · B = |A||B|cos(θ).

A. 60°
B. 45°
C. 30°
D. 90°

Solution

A · B = 1*2 + 2*3 = 8. |A| = √(1^2 + 2^2) = √5, |B| = √(2^2 + 3^2) = √13. cos(θ) = 8/(√5*√13).

Correct Answer: B — 45°

Q. Find the angle θ between vectors A = 4i + 3j and B = 1i + 2j if A · B = |A||B|cos(θ).

A. 60°
B. 45°
C. 30°
D. 90°

Solution

A · B = 4*1 + 3*2 = 10; |A| = √(4^2 + 3^2) = 5; |B| = √(1^2 + 2^2) = √5; cos(θ) = 10/(5√5) = 2/√5; θ = 45°.

Correct Answer: B — 45°

Q. Find the scalar product of A = 2i + 3j + k and B = i + 2j + 3k. (2020)

A. 14
B. 12
C. 10
D. 8

Solution

A · B = (2)(1) + (3)(2) + (1)(3) = 2 + 6 + 3 = 11

Correct Answer: A — 14

Q. Find the scalar product of A = 6i + 8j and B = 2i + 3j.

A. 42
B. 54
C. 48
D. 36

Solution

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

Correct Answer: A — 42

Q. Find the scalar product of the vectors A = 7i - 2j + k and B = 3i + 4j - 5k.

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

Solution

A · B = (7)(3) + (-2)(4) + (1)(-5) = 21 - 8 - 5 = 8.

Correct Answer: A — -1

Q. Find the scalar product of vectors A = 7i + 1j + 2k and B = 3i + 4j + 5k.

A. 43
B. 37
C. 35
D. 41

Solution

A · B = (7)(3) + (1)(4) + (2)(5) = 21 + 4 + 10 = 35.

Correct Answer: A — 43

Q. Find the unit vector in the direction of vector A = 6i - 8j.

A. 3/5 i - 4/5 j
B. 6/10 i - 8/10 j
C. 1/5 i - 2/5 j
D. 2/5 i - 3/5 j

Solution

Magnitude |A| = √(6^2 + (-8)^2) = √(36 + 64) = 10. Unit vector = (6/10)i + (-8/10)j = (3/5)i - (4/5)j.

Correct Answer: A — 3/5 i - 4/5 j

Q. Find the unit vector in the direction of vector D = -3i + 4j.

A. -0.6i + 0.8j
B. 0.6i - 0.8j
C. 0.8i + 0.6j
D. -0.8i + 0.6j

Solution

Magnitude |D| = √((-3)^2 + 4^2) = √(9 + 16) = √25 = 5. Unit vector = D/|D| = (-3/5)i + (4/5)j = -0.6i + 0.8j.

Correct Answer: A — -0.6i + 0.8j

Q. For vectors A = 2i + 3j and B = 5i + 6j, what is A · B?

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

Solution

A · B = (2)(5) + (3)(6) = 10 + 18 = 28.

Correct Answer: A — 28

Q. For vectors A = 2i + j and B = 3i + 4j, what is the scalar product A · B?

A. 14
B. 10
C. 12
D. 8

Solution

A · B = (2)(3) + (1)(4) = 6 + 4 = 10.

Correct Answer: A — 14

Q. For vectors A = 4i + 3j and B = 3i - 4j, find A · B.

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

Solution

A · B = (4)(3) + (3)(-4) = 12 - 12 = 0.

Correct Answer: A — -6

Q. For vectors A = 6i + 8j and B = 2i + 3j, find A · B.

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

Solution

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

Correct Answer: A — 42

Q. Given A = 1i + 1j and B = 1i + 1j, what is A · B?

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

Solution

A · B = (1)(1) + (1)(1) = 1 + 1 = 2.

Correct Answer: A — 2

Q. Given A = 2i + 2j and B = 3i + 3j, what is the scalar product A · B?

A. 12
B. 18
C. 10
D. 14

Solution

A · B = (2)(3) + (2)(3) = 6 + 6 = 12.

Correct Answer: A — 12

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

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

Solution

A · B = (2)(3) + (2)(4) = 6 + 8 = 14.

Correct Answer: A — 14

Q. Given vectors A = 4i + 2j and B = -i + 3j, calculate A · B.

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

Solution

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

Correct Answer: C — 8

Q. Given vectors A = 4i + 3j and B = 1i + 2j, calculate A · B.

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

Solution

A · B = (4)(1) + (3)(2) = 4 + 6 = 10.

Correct Answer: A — 10

Q. Given vectors A = i + 2j + 3k and B = 2i + 3j + 4k, calculate A · B.

A. 20
B. 22
C. 24
D. 26

Solution

A · B = (1)(2) + (2)(3) + (3)(4) = 2 + 6 + 12 = 20.

Correct Answer: C — 24

Q. Given vectors A = i + 2j + 3k and B = 4i + 5j + 6k, calculate A · B.

A. 32
B. 30
C. 28
D. 34

Solution

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

Correct Answer: B — 30

Q. Given vectors A = i + 2j + 3k and B = 4i + 5j + 6k, find A · B. (2019)

A. 32
B. 34
C. 36
D. 38

Solution

A · B = (1)(4) + (2)(5) + (3)(6) = 4 + 10 + 18 = 32

Correct Answer: B — 34

Q. Given vectors A = i + 2j and B = 3i + 4j, calculate A · B. (2022)

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

Solution

A · B = (1)(3) + (2)(4) = 3 + 8 = 11.

Correct Answer: A — 11

Q. If A = 1i + 1j + 1k and B = 1i + 1j + 1k, what is A · B?

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

Solution

A · B = (1)(1) + (1)(1) + (1)(1) = 1 + 1 + 1 = 3.

Correct Answer: C — 3

Q. If A = 1i + 1j + 1k and B = 1i + 2j + 3k, find A · B.

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

Solution

A · B = (1)(1) + (1)(2) + (1)(3) = 1 + 2 + 3 = 6.

Correct Answer: A — 6

Q. If A = 1i + 1j + 1k and B = 1i + 2j + 3k, what is A · B?

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

Solution

A · B = (1)(1) + (1)(2) + (1)(3) = 1 + 2 + 3 = 6.

Correct Answer: B — 5

Q. If A = 1i + 1j + 1k and B = 2i + 2j + 2k, find A · B.

A. 6
B. 4
C. 2
D. 8

Solution

A · B = (1)(2) + (1)(2) + (1)(2) = 2 + 2 + 2 = 6.

Correct Answer: A — 6

Q. If A = 1i + 1j and B = 1i - 1j, what is the scalar product A · B?

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

Solution

A · B = (1)(1) + (1)(-1) = 1 - 1 = 0.

Correct Answer: A — 0

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