Q. If the vector a = (2, 2) is scaled by a factor of 3, what is the resulting vector?
A.
(6, 6)
B.
(3, 3)
C.
(2, 2)
D.
(1, 1)
Show solution
Solution
Scaled vector = 3 * a = 3 * (2, 2) = (6, 6)
Correct Answer:
A
— (6, 6)
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Q. If the vector a = (2, 3) and b = (4, 1), what is the cross product a × b?
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Solution
Cross product a × b = 2*1 - 3*4 = 2 - 12 = -10.
Correct Answer:
A
— -10
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Q. If the vector a = (2, 3) and b = (4, 1), what is the resultant vector a + b?
A.
(6, 4)
B.
(2, 4)
C.
(4, 2)
D.
(6, 2)
Show solution
Solution
Resultant vector a + b = (2+4, 3+1) = (6, 4).
Correct Answer:
A
— (6, 4)
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Q. If the vector A = (2, 3) is multiplied by 2, what is the resulting vector?
A.
(4, 6)
B.
(2, 3)
C.
(1, 1.5)
D.
(0, 0)
Show solution
Solution
Resulting vector = 2 * A = 2 * (2, 3) = (4, 6).
Correct Answer:
A
— (4, 6)
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Q. If the vector A = (2, 3) is reflected across the line y = x, what is the resulting vector?
A.
(3, 2)
B.
(2, 3)
C.
(0, 0)
D.
(1, 1)
Show solution
Solution
Reflection across y = x gives vector (3, 2).
Correct Answer:
A
— (3, 2)
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Q. If the vector A = (2, 3) is scaled by a factor of 2, what is the resulting vector?
A.
(4, 6)
B.
(2, 3)
C.
(1, 1.5)
D.
(0, 0)
Show solution
Solution
Scaled vector = 2 * A = 2 * (2, 3) = (4, 6).
Correct Answer:
A
— (4, 6)
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Q. If the vector a = (2, 3, 4) and b = (1, 0, -1), what is a + b?
A.
(3, 3, 3)
B.
(1, 3, 3)
C.
(2, 3, 3)
D.
(2, 3, 5)
Show solution
Solution
a + b = (2+1, 3+0, 4-1) = (3, 3, 3).
Correct Answer:
A
— (3, 3, 3)
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Q. If the vector a = (2, 3, 4) and b = (1, 0, -1), what is the scalar triple product a · (b × a)?
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Solution
The scalar triple product is 0 because a · (b × a) = 0.
Correct Answer:
A
— 0
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Q. If the vector a = (2, 3, 4) is scaled by a factor of 2, what is the resulting vector?
A.
(4, 6, 8)
B.
(2, 3, 4)
C.
(1, 1.5, 2)
D.
(0, 0, 0)
Show solution
Solution
Scaling the vector a by 2 gives (2*2, 2*3, 2*4) = (4, 6, 8).
Correct Answer:
A
— (4, 6, 8)
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Q. If the vector a = (3, 4) and b = (1, 2), find the cross product a × b.
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Solution
In 2D, a × b = a1*b2 - a2*b1 = 3*2 - 4*1 = 6 - 4 = 2.
Correct Answer:
A
— -2
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Q. If the vector a = (3, 4) is scaled by a factor of 2, what is the new vector?
A.
(6, 8)
B.
(3, 4)
C.
(1.5, 2)
D.
(0, 0)
Show solution
Solution
New vector = 2 * (3, 4) = (6, 8).
Correct Answer:
A
— (6, 8)
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Q. If the vector a = (3, 4, 0) and b = (0, 0, 5), what is the magnitude of a × b?
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Solution
Magnitude of a × b = |a||b|sin(90) = |(3, 4, 0)|| (0, 0, 5)| = 5√(3^2 + 4^2) = 15.
Correct Answer:
A
— 15
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Q. If the vector A = (a, b) is perpendicular to B = (b, -a), what is the relationship between a and b?
A.
a = b
B.
a = -b
C.
a + b = 0
D.
a - b = 0
Show solution
Solution
A·B = ab - ab = 0, hence A and B are perpendicular if a = -b.
Correct Answer:
B
— a = -b
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Q. If the vectors A = (1, 2) and B = (2, 1) are given, what is the angle between them?
A.
90 degrees
B.
45 degrees
C.
60 degrees
D.
30 degrees
Show solution
Solution
Cosine of angle θ = (A · B) / (|A| |B|) = (1*2 + 2*1) / (√5 * √5) = 4/5, θ = cos⁻¹(4/5).
Correct Answer:
B
— 45 degrees
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Q. If u = (1, 2) and v = (3, 4), what is the dot product u · v?
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Solution
Dot product u · v = 1*3 + 2*4 = 3 + 8 = 11.
Correct Answer:
A
— 10
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Q. If u = (1, 2) and v = (3, 4), what is u + v?
A.
(4, 6)
B.
(2, 3)
C.
(1, 2)
D.
(3, 4)
Show solution
Solution
u + v = (1 + 3, 2 + 4) = (4, 6)
Correct Answer:
A
— (4, 6)
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Q. If u = (1, 2, 3) and v = (4, 5, 6), what is the dot product u · v?
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Solution
Dot product u · v = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.
Correct Answer:
B
— 27
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Q. If u = (2, 3, 1) and v = (1, 0, -1), find the dot product u · v.
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Solution
u · v = 2*1 + 3*0 + 1*(-1) = 2 + 0 - 1 = 1.
Correct Answer:
A
— 5
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Q. If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is A + B?
A.
(5, 7, 9)
B.
(4, 5, 6)
C.
(1, 2, 3)
D.
(0, 0, 0)
Show solution
Solution
A + B = (1+4, 2+5, 3+6) = (5, 7, 9).
Correct Answer:
A
— (5, 7, 9)
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Q. If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is the angle between them?
A.
0 degrees
B.
30 degrees
C.
60 degrees
D.
90 degrees
Show solution
Solution
Cosine of angle θ = (A . B) / (|A| |B|) = (1*4 + 2*5 + 3*6) / (√14 * √77) = 0, hence θ = 90 degrees.
Correct Answer:
D
— 90 degrees
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Q. If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is the vector A - B?
A.
(-3, -3, -3)
B.
(3, 3, 3)
C.
(5, 7, 9)
D.
(0, 0, 0)
Show solution
Solution
A - B = (1-4, 2-5, 3-6) = (-3, -3, -3).
Correct Answer:
A
— (-3, -3, -3)
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Q. If vector A = (2, 2, 2) and vector B = (1, 1, 1), what is the scalar triple product A . (B × A)?
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Solution
A . (B × A) = 0, since B × A = 0.
Correct Answer:
A
— 0
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Q. If vector A = (3, -2, 1) and vector B = (1, 4, -3), what is the cross product A × B?
A.
(-5, -10, 14)
B.
(5, 10, -14)
C.
(10, 14, 5)
D.
(14, -5, 10)
Show solution
Solution
A × B = |i j k|\n|3 -2 1|\n|1 4 -3| = (-5, -10, 14).
Correct Answer:
A
— (-5, -10, 14)
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Q. What is the angle between the vectors (1, 0) and (0, 1)?
A.
0 degrees
B.
90 degrees
C.
45 degrees
D.
180 degrees
Show solution
Solution
The angle between (1, 0) and (0, 1) is 90 degrees.
Correct Answer:
B
— 90 degrees
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Q. What is the angle between the vectors (1, 2, 2) and (2, 1, 2)?
A.
90 degrees
B.
60 degrees
C.
45 degrees
D.
30 degrees
Show solution
Solution
Cosine of angle θ = (u · v) / (|u| |v|). Calculate to find θ = 60 degrees.
Correct Answer:
B
— 60 degrees
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Q. What is the angle between the vectors a = (1, 2, 2) and b = (2, 0, 2)?
A.
0 degrees
B.
45 degrees
C.
90 degrees
D.
60 degrees
Show solution
Solution
cos(θ) = (a · b) / (|a| |b|). Calculate 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. Thus, cos(θ) = 6 / (3 * 2√2) = 1/√2, θ = 45 degrees.
Correct Answer:
D
— 60 degrees
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Q. What is the angle between the vectors u = (1, 0) and v = (0, 1)?
A.
0 degrees
B.
90 degrees
C.
45 degrees
D.
180 degrees
Show solution
Solution
The angle between u and v is 90 degrees since they are perpendicular.
Correct Answer:
B
— 90 degrees
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Q. What is the angle between vectors A = (1, 0, 0) and B = (0, 1, 0)?
A.
0 degrees
B.
45 degrees
C.
90 degrees
D.
180 degrees
Show solution
Solution
The angle θ = cos⁻¹((A . B) / (|A| |B|)) = cos⁻¹(0) = 90 degrees.
Correct Answer:
C
— 90 degrees
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Q. What is the cross product of the vectors (1, 0, 0) and (0, 1, 0)?
A.
(0, 0, 1)
B.
(1, 1, 0)
C.
(0, 0, 0)
D.
(1, 0, 0)
Show solution
Solution
Cross product = (1, 0, 0) × (0, 1, 0) = (0, 0, 1).
Correct Answer:
A
— (0, 0, 1)
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Q. What is the cross product of the vectors (1, 2, 3) and (4, 5, 6)?
A.
(-3, 6, -3)
B.
(-3, 6, 3)
C.
(3, -6, 3)
D.
(3, 6, -3)
Show solution
Solution
Cross product = |i j k| |1 2 3| |4 5 6| = (-3, 6, -3).
Correct Answer:
A
— (-3, 6, -3)
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Showing 31 to 60 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!
