Vectors in 2D & 3D for Competitive Exam Preparation

Vectors in 2D & 3D

Q. If vector A = 6i + 8j, what is the unit vector in the direction of A? (2023)

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) = 10. Unit vector = (1/10)(6i + 8j) = (3/5)i + (4/5)j.

Correct Answer: A — 3/5 i + 4/5 j

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Q. If vector G = 4i - 3j + 2k, what is the y-component of G?

A. 4
B. -3
C. 2
D. 0

Solution

The y-component of G is -3.

Correct Answer: B — -3

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Q. If vector J = 5i + 12j, what is the angle between J and the positive x-axis?

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

Solution

tan(θ) = 12/5; θ = tan^(-1)(12/5) which is approximately 67.38 degrees, closest to 60 degrees.

Correct Answer: C — 60 degrees

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Q. If vector K = 2i + 2j and vector L = -i + 3j, what is the resultant vector K + L?

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

Solution

K + L = (2i + 2j) + (-i + 3j) = (2 - 1)i + (2 + 3)j = 1i + 5j = i + 5j.

Correct Answer: A — i + 5j

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Q. What is the angle between the vectors A = 2i + 2j and B = 2i - 2j?

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

Solution

cos(θ) = (A · B) / (|A||B|) = (8) / (√8 * √8) = 1, θ = 0 degrees.

Correct Answer: C — 90 degrees

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Q. What is the angle between the vectors A = i + j and B = 2i + 2j?

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

Solution

cos(θ) = (A · B) / (|A||B|) = (1*2 + 1*2) / (√2 * √8) = 4 / (2√2) = √2. θ = 45 degrees.

Correct Answer: A — 45 degrees

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Q. What is the cross product of vectors A = i + 2j + 3k and B = 4i + 5j + 6k?

A. -3i + 6j - 3k
B. -3i + 6j + 3k
C. 3i - 6j + 3k
D. 3i + 6j - 3k

Solution

A × B = |i  j  k| |1  2  3| |4  5  6| = -3i + 6j - 3k.

Correct Answer: A — -3i + 6j - 3k

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Q. What is the cross product of vectors A = i + 2j and B = 3i + 4j? (2021)

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

Solution

A × B = |i  j  k| |1  2  0| |3  4  0| = (0 - 0)i - (0 - 0)j + (4 - 6)k = -2k.

Correct Answer: A — -2k

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Q. What is the cross product of vectors E = i + 2j and F = 3i + 4j?

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

Solution

E × F = |i  j  k| |1  2  0| |3  4  0| = (0 - 0)i - (0 - 0)j + (1*4 - 2*3)k = -2k.

Correct Answer: A — -2k

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Q. What is the magnitude of the vector C = 5i - 12j?

A. 13
B. 12
C. 5
D. 17

Solution

Magnitude |C| = √(5^2 + (-12)^2) = √(25 + 144) = √169 = 13.

Correct Answer: A — 13

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Q. What is the projection of vector A = 3i + 4j onto vector B = 1i + 2j?

A. 2.5
B. 3
C. 4
D. 5

Solution

Projection = (A · B / |B|^2) * B = (11 / 5) * (1i + 2j) = 2.5.

Correct Answer: A — 2.5

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Q. What is the projection of vector A = 3i + 4j onto vector B = 2i + 0j?

A. 6
B. 3
C. 4
D. 0

Solution

Projection of A onto B = (A · B / |B|^2)B. A · B = 6, |B|^2 = 4. Projection = (6/4)(2i) = 3i.

Correct Answer: A — 6

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Q. What is the projection of vector A = 3i + 4j onto vector B = 2i + 2j?

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

Solution

Projection = (A · B / |B|^2) * B = (14 / 8) * (2i + 2j) = (7/4)(2i + 2j) = 3.5i + 3.5j.

Correct Answer: A — 5

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Q. What is the projection of vector A = 6i + 8j onto vector B = 2i + 2j?

A. 8
B. 6
C. 4
D. 10

Solution

Projection of A onto B = (A · B / |B|^2) * B = ((6*2 + 8*2) / (2^2 + 2^2)) * (2i + 2j) = (28/8)(2i + 2j) = 7i + 7j.

Correct Answer: A — 8

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Q. What is the scalar projection of vector H = 6i + 8j onto vector I = 3i + 4j?

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

Solution

Scalar projection = (H · I) / |I| = (6*3 + 8*4) / √(3^2 + 4^2) = (18 + 32) / 5 = 50 / 5 = 10.

Correct Answer: C — 10

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Q. What is the scalar triple product of vectors A = i + j + k, B = 2i + 3j + k, and C = 3i + j + 2k?

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

Solution

Scalar triple product = A · (B × C) = |i  j  k| |1  1  1| |2  3  1| |3  1  2| = 0.

Correct Answer: D — 0

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Q. What is the unit vector in the direction of vector A = 4i + 3j?

A. (4/5)i + (3/5)j
B. (3/4)i + (4/3)j
C. (4/3)i + (3/4)j
D. (3/5)i + (4/5)j

Solution

Unit vector = A / |A| = (4i + 3j) / √(4^2 + 3^2) = (4i + 3j) / 5 = (4/5)i + (3/5)j.

Correct Answer: A — (4/5)i + (3/5)j

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Q. Which of the following vectors is orthogonal to the vector A = 2i + 3j?

A. 3i - 2j
B. -3i + 2j
C. 2i + 3j
D. i + j

Solution

A vector is orthogonal if A · B = 0. For B = 3i - 2j, A · B = 6 - 6 = 0.

Correct Answer: A — 3i - 2j

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Showing 31 to 48 of 48 (2 Pages)

Vectors in 2D & 3D MCQ & Objective Questions

Understanding "Vectors in 2D & 3D" is crucial for students preparing for school and competitive exams. This topic not only forms the foundation of various mathematical concepts but also plays a significant role in physics and engineering. Practicing MCQs and objective questions on vectors helps students enhance their problem-solving skills and boosts their confidence, leading to better scores in exams.

What You Will Practise Here

Understanding the definition and properties of vectors.
Operations on vectors: addition, subtraction, and scalar multiplication.
Vector representation in 2D and 3D coordinate systems.
Dot product and cross product of vectors with applications.
Magnitude and direction of vectors, including unit vectors.
Applications of vectors in real-life scenarios and physics problems.
Graphical representation of vectors using diagrams.

Exam Relevance

The topic of vectors is frequently covered in CBSE, State Boards, NEET, and JEE exams. Students can expect questions that assess their understanding of vector operations, graphical representation, and applications. Common question patterns include numerical problems, conceptual MCQs, and theoretical questions that require a clear grasp of vector properties and their applications.

Common Mistakes Students Make

Confusing the dot product and cross product and their respective applications.
Misunderstanding the concept of vector magnitude and direction.
Errors in graphical representation and interpretation of vectors.
Inadequate practice leading to difficulty in solving complex vector problems.

FAQs

Question: What are vectors in 2D and 3D?
Answer: Vectors are quantities that have both magnitude and direction. In 2D, they are represented in a plane, while in 3D, they extend into space.

Question: How do I calculate the dot product of two vectors?
Answer: The dot product is calculated by multiplying the corresponding components of the vectors and summing the results.

Question: Why are vectors important in physics?
Answer: Vectors are essential in physics as they help describe forces, velocities, and other quantities that have direction and magnitude.

Ready to master "Vectors in 2D & 3D"? Dive into our practice MCQs and test your understanding to excel in your exams!

Vectors in 2D & 3D

Vectors in 2D & 3D MCQ & Objective Questions

What You Will Practise Here

Exam Relevance

Common Mistakes Students Make

FAQs

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