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?
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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 scalar product of two vectors A and B is 0, what can be said about the vectors?
A.
They are parallel
B.
They are orthogonal
C.
They are equal
D.
They are collinear
Show solution
Solution
If A · B = 0, then the vectors are orthogonal.
Correct Answer:
B
— They are orthogonal
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Q. If the scalar product of vectors A = (x, y, z) and B = (2, -1, 3) is 10, what is the equation?
A.
2x - y + 3z = 10
B.
2x + y + 3z = 10
C.
2x - y - 3z = 10
D.
2x + y - 3z = 10
Show solution
Solution
A · B = 2x - y + 3z = 10.
Correct Answer:
A
— 2x - y + 3z = 10
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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?
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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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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?
Show solution
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 the vectors A = (2, 3) and B = (4, 5) are given, what is the scalar product A · B?
Show solution
Solution
A · B = 2*4 + 3*5 = 8 + 15 = 23.
Correct Answer:
C
— 20
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Q. If the vectors A = (3, -2, 1) and B = (k, 4, -2) are orthogonal, find the value of k.
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Solution
A · B = 3k - 8 - 2 = 0; 3k - 10 = 0; k = 10/3.
Correct Answer:
A
— -1
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Q. If the vectors A = (x, 2, 3) and B = (4, y, 6) are orthogonal, what is the value of y?
Show solution
Solution
A · B = x*4 + 2*y + 3*6 = 0. Thus, 4x + 2y + 18 = 0. If x = 0, then y = -9. If x = 1, y = -10. The only integer solution is y = 3.
Correct Answer:
B
— 3
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Showing 121 to 150 of 210 (7 Pages)
Vector & 3D Geometry MCQ & Objective Questions
Understanding Vector & 3D Geometry is crucial for students preparing for various school and competitive exams. This topic not only enhances spatial reasoning but also forms the backbone of many important concepts in mathematics and physics. Practicing MCQs and objective questions in this area can significantly improve your exam scores and boost your confidence. Engaging with practice questions helps solidify your grasp of key concepts and prepares you for tackling important questions effectively.
What You Will Practise Here
Basics of vectors: definitions, types, and operations
Vector addition and subtraction: graphical and algebraic methods
Dot product and cross product: properties and applications
Equations of lines and planes in 3D space
Distance between points, lines, and planes
Applications of vectors in physics: force, velocity, and acceleration
Common theorems and formulas related to 3D geometry
Exam Relevance
Vector & 3D Geometry is a significant topic in CBSE, State Boards, NEET, and JEE exams. Students can expect questions that assess their understanding of vector operations, geometric interpretations, and problem-solving skills. Common question patterns include multiple-choice questions that require students to apply concepts to real-world scenarios, as well as numerical problems that test their computational abilities.
Common Mistakes Students Make
Confusing the dot product and cross product, leading to incorrect applications.
Misinterpreting the geometric representation of vectors, especially in 3D space.
Overlooking the significance of direction in vector addition and subtraction.
Failing to apply the correct formulas for distance calculations between geometric entities.
FAQs
Question: What are the key formulas I should remember for Vector & 3D Geometry?Answer: Important formulas include the dot product formula, cross product formula, and distance formulas between points, lines, and planes.
Question: How can I improve my understanding of Vector & 3D Geometry concepts?Answer: Regular practice of MCQs and solving objective questions will help reinforce your understanding and application of these concepts.
Start your journey towards mastering Vector & 3D Geometry today! Solve practice MCQs to test your understanding and enhance your exam preparation. Your success is just a question away!
