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
Show solution
Solution
The angle θ = cos⁻¹((u · v) / (|u| |v|)) = cos⁻¹(0) = 90 degrees.
Correct Answer:
B
— 90 degrees
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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)
Show solution
Solution
Cross product A × B = |i j k| |1 2 3| |4 5 6| = (-3, 6, -3).
Correct Answer:
A
— (-3, 6, -3)
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Q. Find the magnitude of the vector (3, 4).
Show solution
Solution
Magnitude = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.
Correct Answer:
A
— 5
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Q. Find the magnitude of the vector v = (3, -4, 12).
Show solution
Solution
Magnitude |v| = √(3^2 + (-4)^2 + 12^2) = √(9 + 16 + 144) = √169 = 13.
Correct Answer:
B
— 14
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Q. Find the scalar projection of vector A = (3, 4) onto vector B = (1, 0).
Show solution
Solution
Scalar projection = (A · B) / |B| = (3*1 + 4*0) / 1 = 3.
Correct Answer:
A
— 3
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Q. Find the unit vector in the direction of the vector (3, 4).
A.
(0.6, 0.8)
B.
(0.8, 0.6)
C.
(1, 1)
D.
(0.5, 0.5)
Show solution
Solution
Magnitude = √(3^2 + 4^2) = 5. Unit vector = (3/5, 4/5) = (0.6, 0.8).
Correct Answer:
A
— (0.6, 0.8)
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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)
Show solution
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)
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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)
Show solution
Solution
Unit vector = (4, 3) / √(4^2 + 3^2) = (4, 3) / 5 = (4/5, 3/5).
Correct Answer:
A
— (4/5, 3/5)
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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)
Show solution
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)
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Q. Find the unit vector in the direction of the vector v = (4, -3).
A.
(4/5, -3/5)
B.
(3/5, 4/5)
C.
(4/3, -3/4)
D.
(3/4, 4/3)
Show solution
Solution
Magnitude |v| = √(4^2 + (-3)^2) = √(16 + 9) = 5. Unit vector = (4/5, -3/5).
Correct Answer:
A
— (4/5, -3/5)
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Q. If A = (1, 0) and B = (0, 1), what is the angle between them?
A.
0 degrees
B.
90 degrees
C.
45 degrees
D.
180 degrees
Show solution
Solution
Angle = cos⁻¹((A·B) / (|A||B|)) = cos⁻¹(0) = 90 degrees
Correct Answer:
B
— 90 degrees
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Q. If A = (1, 2) and B = (3, 4), what is the dot product A · B?
Show solution
Solution
Dot product A · B = 1*3 + 2*4 = 3 + 8 = 11.
Correct Answer:
A
— 10
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Q. If A = (1, 2) and B = (3, 4), what is the midpoint M of AB?
A.
(2, 3)
B.
(1, 2)
C.
(3, 4)
D.
(4, 5)
Show solution
Solution
Midpoint M = ((1+3)/2, (2+4)/2) = (2, 3).
Correct Answer:
A
— (2, 3)
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Q. If A = (2, 3) and B = (4, 5), what is the vector AB?
A.
(2, 2)
B.
(2, 3)
C.
(4, 5)
D.
(6, 8)
Show solution
Solution
AB = B - A = (4 - 2, 5 - 3) = (2, 2)
Correct Answer:
A
— (2, 2)
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Q. If A = (2, 3) and B = (4, 7), find the vector AB.
A.
(2, 4)
B.
(2, 3)
C.
(2, 1)
D.
(2, 2)
Show solution
Solution
Vector AB = B - A = (4 - 2, 7 - 3) = (2, 4).
Correct Answer:
A
— (2, 4)
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Q. If A(1, 2, 3) and B(4, 5, 6) are two points in space, what is the vector AB?
A.
(3, 3, 3)
B.
(2, 3, 4)
C.
(1, 1, 1)
D.
(0, 0, 0)
Show solution
Solution
Vector AB = B - A = (4-1, 5-2, 6-3) = (3, 3, 3).
Correct Answer:
A
— (3, 3, 3)
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Q. If A(1, 2, 3) and B(4, 5, 6) are two points, what is the vector AB?
A.
(3, 3, 3)
B.
(3, 3, 0)
C.
(0, 0, 0)
D.
(1, 1, 1)
Show solution
Solution
Vector AB = B - A = (4-1, 5-2, 6-3) = (3, 3, 3).
Correct Answer:
A
— (3, 3, 3)
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Q. If A(2, 3, 4) and B(1, 0, -1) are two points in space, find the vector AB.
A.
(1, 3, 5)
B.
(1, -3, -5)
C.
(1, 3, -5)
D.
(1, -3, 5)
Show solution
Solution
AB = B - A = (1 - 2, 0 - 3, -1 - 4) = (-1, -3, -5) = (1, 3, 5) in the opposite direction.
Correct Answer:
A
— (1, 3, 5)
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Q. If the position vector of a point is (5, 12), what is its distance from the origin?
Show solution
Solution
Distance = √(5^2 + 12^2) = √(25 + 144) = √169 = 13
Correct Answer:
A
— 13
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Q. If the position vector of a point is given by r = (2t, 3t, 4t), what is the velocity vector?
A.
(2, 3, 4)
B.
(4, 6, 8)
C.
(2t, 3t, 4t)
D.
(0, 0, 0)
Show solution
Solution
Velocity vector = dr/dt = (2, 3, 4)
Correct Answer:
A
— (2, 3, 4)
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Q. If the position vector of a point P is (2, 3, 4), what is the distance from the origin to point P?
Show solution
Solution
Distance = √(2^2 + 3^2 + 4^2) = √(4 + 9 + 16) = √29 ≈ 5.385.
Correct Answer:
B
— 6
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Q. If the position vector of a point P is (x, y, z) and the vector a = (1, 2, 3), what is the projection of P onto a?
A.
(1, 2, 3)
B.
(2, 4, 6)
C.
(0, 0, 0)
D.
(x, y, z)
Show solution
Solution
Projection of P onto a = ((P · a) / |a|^2) * a.
Correct Answer:
D
— (x, y, z)
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Q. If the position vector of a point P is given by r = (2t, 3t, 4t), find the coordinates of P when t = 1.
A.
(2, 3, 4)
B.
(1, 1, 1)
C.
(0, 0, 0)
D.
(2, 4, 6)
Show solution
Solution
Substituting t = 1, r = (2*1, 3*1, 4*1) = (2, 3, 4).
Correct Answer:
A
— (2, 3, 4)
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Q. If the position vector of point P is (3, -2) and Q is (1, 4), what is the vector PQ?
A.
(-2, 6)
B.
(2, -6)
C.
(4, -6)
D.
(6, 2)
Show solution
Solution
Vector PQ = Q - P = (1 - 3, 4 - (-2)) = (-2, 6).
Correct Answer:
A
— (-2, 6)
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Q. If the position vector of point P is (3, 4) and Q is (1, 2), what is the vector PQ?
A.
(2, 2)
B.
(4, 6)
C.
(2, 4)
D.
(1, 1)
Show solution
Solution
Vector PQ = Q - P = (1 - 3, 2 - 4) = (-2, -2).
Correct Answer:
A
— (2, 2)
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Q. If the vector A = (1, 2) and B = (2, 1), what is the angle between them?
A.
0 degrees
B.
90 degrees
C.
45 degrees
D.
180 degrees
Show solution
Solution
Cosine of angle = (A · B) / (|A| |B|) = (1*2 + 2*1) / (√5 * √5) = 4/5, angle = cos^(-1)(4/5).
Correct Answer:
C
— 45 degrees
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Q. If the vector a = (1, 2) and b = (3, 4), find the angle between them using the dot product.
A.
0 degrees
B.
90 degrees
C.
45 degrees
D.
60 degrees
Show solution
Solution
cos(θ) = (a · b) / (|a| |b|). a · b = 1*3 + 2*4 = 11, |a| = √(1^2 + 2^2) = √5, |b| = √(3^2 + 4^2) = 5. Thus, cos(θ) = 11 / (√5 * 5) = 11 / (5√5), θ = 60 degrees.
Correct Answer:
D
— 60 degrees
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Q. If the vector a = (2, -1) and b = (1, 3), what is a + b?
A.
(3, 2)
B.
(1, 2)
C.
(2, 2)
D.
(3, 1)
Show solution
Solution
a + b = (2 + 1, -1 + 3) = (3, 2)
Correct Answer:
A
— (3, 2)
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Q. If the vector a = (2, -1) and b = (1, 3), what is the cross product a × b?
Show solution
Solution
Cross product in 2D = a1*b2 - a2*b1 = 2*3 - (-1)*1 = 6 + 1 = 7
Correct Answer:
A
— 5
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Q. If the vector a = (2, 2) and b = (2, -2), what is the angle between them?
A.
90 degrees
B.
45 degrees
C.
0 degrees
D.
180 degrees
Show solution
Solution
Angle = cos⁻¹((a·b) / (|a||b|)) = cos⁻¹(0) = 90 degrees
Correct Answer:
A
— 90 degrees
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Showing 1 to 30 of 86 (3 Pages)
Vector Algebra Basics MCQ & Objective Questions
Understanding the fundamentals of Vector Algebra is crucial for students preparing for various exams. Mastering these basics not only enhances conceptual clarity but also significantly boosts your performance in objective questions. Practicing MCQs related to Vector Algebra Basics helps you identify important questions and strengthens your exam preparation strategy.
What You Will Practise Here
Definition and properties of vectors
Vector addition and subtraction
Scalar and vector products
Applications of vectors in geometry
Unit vectors and their significance
Representation of vectors in different dimensions
Key formulas related to vector operations
Exam Relevance
Vector Algebra is a significant topic in various educational boards, including CBSE and State Boards, as well as competitive exams like NEET and JEE. Questions often focus on vector operations, properties, and applications. Common patterns include solving problems using vector addition or finding the angle between vectors, making it essential to grasp these concepts thoroughly.
Common Mistakes Students Make
Confusing scalar and vector quantities
Incorrectly applying vector addition rules
Misunderstanding the concept of unit vectors
Neglecting the geometric interpretation of vectors
Overlooking the importance of direction in vector problems
FAQs
Question: What are the basic operations of vectors?Answer: The basic operations include vector addition, subtraction, and multiplication (both scalar and vector products).
Question: How can I improve my understanding of Vector Algebra?Answer: Regular practice of MCQs and solving objective questions can significantly enhance your understanding and retention of Vector Algebra concepts.
Start solving practice MCQs on Vector Algebra Basics today to test your understanding and prepare effectively for your exams. Remember, consistent practice is the key to success!
