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
Show solution
Solution
cos(θ) = (A · B) / (|A||B|). A · B = 0, hence θ = 90 degrees.
Correct Answer:
C
— 90 degrees
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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
Show solution
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
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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
Show solution
Solution
cos(θ) = (A · B) / (|A||B|) = (1 - 1) / (√2 * √2) = 0, θ = 90 degrees.
Correct Answer:
C
— 90 degrees
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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
Show solution
Solution
cos(θ) = (A · B) / (|A| |B|). A · B = 0, hence θ = 90 degrees.
Correct Answer:
A
— 90 degrees
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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
Show solution
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
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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
Show solution
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
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Q. If the position vector of point P is given by r = 2i + 3j + 4k, what is the distance from the origin to point P?
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Solution
Distance = |r| = √(2^2 + 3^2 + 4^2) = √(4 + 9 + 16) = √29.
Correct Answer:
B
— 6
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Q. If the position vector of point P is given by r = 2i + 3j + 4k, what is the x-component of r?
Show solution
Solution
The x-component of r is 2.
Correct Answer:
A
— 2
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Q. If the position vector of point P is given by r = 2i + 3j + 4k, what is the x-coordinate of point P? (2020)
Show solution
Solution
The x-coordinate of point P is the coefficient of i in the position vector, which is 2.
Correct Answer:
A
— 2
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Q. If the position vector of point P is given by r = 3i + 4j, what is the distance of point P from the origin?
Show solution
Solution
Distance = |r| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.
Correct Answer:
A
— 5
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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
Show solution
Solution
R = A + B = (2 - 1)i + (3 + 4)j = 1i + 7j.
Correct Answer:
A
— i + 7j
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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
Show solution
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
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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
Show solution
Solution
cos(θ) = (A · B) / (|A||B|) = (8 + 15) / (√(13) * √(41)). θ = 60 degrees.
Correct Answer:
D
— 60 degrees
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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
Show solution
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
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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
Show solution
Solution
Vectors A and B are parallel because B = 2A.
Correct Answer:
A
— Yes
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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
Show solution
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
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Q. If vector A = 3i + 4j and vector B = 2i - j, what is the dot product A · B?
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Solution
A · B = (3)(2) + (4)(-1) = 6 - 4 = 2.
Correct Answer:
A
— 10
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Q. If vector A = 3i + 4j and vector B = 4i + 3j, what is the cross product A × B?
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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
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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
Show solution
Solution
cos(θ) = (A · B) / (|A||B|) = (4*-3 + 3*4) / (5*5) = 0, θ = 90 degrees.
Correct Answer:
A
— 90 degrees
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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
Show solution
Solution
A + B = (4 - 1)i + (3 + 2)j = 3i + 5j.
Correct Answer:
A
— 3i + 5j
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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
Show solution
Solution
R = A + B = (4 - 1)i + (3 + 2)j = 3i + 5j.
Correct Answer:
A
— 3i + 5j
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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
Show solution
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
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Q. If vector A = 4i + 3j and vector B = 4i - 3j, what is the angle between A and B? (2019)
A.
0 degrees
B.
90 degrees
C.
180 degrees
D.
45 degrees
Show solution
Solution
A · B = 16 - 9 = 7. |A| = 5, |B| = 5. cos(θ) = 7/(5*5) = 0.28, θ = 180 degrees.
Correct Answer:
C
— 180 degrees
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Q. If vector A = 5i + 12j and vector B = -5i + 12j, what is the angle between them?
A.
0 degrees
B.
90 degrees
C.
180 degrees
D.
45 degrees
Show solution
Solution
A · B = (5)(-5) + (12)(12) = -25 + 144 = 119. Since A and B are in opposite directions, the angle is 180 degrees.
Correct Answer:
C
— 180 degrees
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Q. If vector A = 5i + 12j and vector B = -5i + 12j, what is the resultant vector A + B? (2021)
A.
0i + 24j
B.
10i + 0j
C.
0i + 12j
D.
5i + 12j
Show solution
Solution
A + B = (5 - 5)i + (12 + 12)j = 0i + 24j.
Correct Answer:
A
— 0i + 24j
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Q. If vector A = 5i + 12j and vector B = 12i - 5j, what is the angle between A and B?
A.
30 degrees
B.
60 degrees
C.
90 degrees
D.
120 degrees
Show solution
Solution
cos(θ) = (A · B) / (|A||B|) = (5*12 + 12*(-5)) / (√(169) * √(169)) = 0, θ = 90 degrees.
Correct Answer:
C
— 90 degrees
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Q. If vector A = 5i + 12j and vector B = 12i - 5j, what is the value of A × B?
Show solution
Solution
A × B = |i j k| |5 12 0| |12 -5 0| = (0 - 0)i - (0 - 0)j + (5*-5 - 12*12)k = -85k.
Correct Answer:
A
— -85
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Q. If vector A = 5i + 12j and vector B = 5i - 12j, what is the angle between A and B?
A.
0 degrees
B.
90 degrees
C.
180 degrees
D.
45 degrees
Show solution
Solution
A · B = 5*5 + 12*(-12) = 25 - 144 = -119. Since A · B < 0, angle is 180 degrees.
Correct Answer:
C
— 180 degrees
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Q. If vector A = 5i + 5j and vector B = 5i - 5j, what is the angle between A and B?
A.
0 degrees
B.
45 degrees
C.
90 degrees
D.
180 degrees
Show solution
Solution
cos(θ) = (A · B) / (|A||B|) = (25 - 25) / (√(50) * √(50)) = 0, θ = 90 degrees.
Correct Answer:
C
— 90 degrees
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Q. If vector A = 5i + 5j and vector B = 5i - 5j, what is the angle between them?
A.
0 degrees
B.
45 degrees
C.
90 degrees
D.
135 degrees
Show solution
Solution
cos(θ) = (A · B) / (|A||B|) = (25 - 25) / (5√2 * 5√2) = 0, θ = 90 degrees.
Correct Answer:
C
— 90 degrees
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Showing 1 to 30 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!
