Vector Algebra: 50 Practice Questions for Competitive Exams
Below are 50 questions on Vector Algebra for UP TGT/PGT, NDA, IAS, and KVS exams. Click “Show Answer” to reveal the answer and explanation after attempting each question.
1. If a = 2i + 3j + k and b = i – j + 2k, find a + b:
a) 3i + 2j + 3k
b) i + 4j – k
c) 3i + 4j + k
d) 2i – j + k
Explanation: a + b = (2i + 3j + k) + (i – j + 2k) = (2+1)i + (3-1)j + (1+2)k = 3i + 2j + 3k.
Year: UP TGT 2016
2. Find the magnitude of vector a = 3i – 4j + 5k:
a) √50
b) 5
c) √34
d) 12
Explanation: |a| = √(3^2 + (-4)^2 + 5^2) = √(9 + 16 + 25) = √50.
Year: KVS PGT 2018
3. If a = 2i + j – k and b = i + 2j + k, find a · b:
a) 3
b) 5
c) 2
d) 4
Explanation: a · b = (2)(1) + (1)(2) + (-1)(1) = 2 + 2 – 1 = 5.
Year: NDA 2019
4. Find the unit vector in the direction of a = 4i – 3j + 12k:
a) (4/13)i – (3/13)j + (12/13)k
b) (4/5)i – (3/5)j + (12/5)k
c) (4/169)i – (3/169)j + (12/169)k
d) (1/13)i – (1/13)j + (1/13)k
Explanation: |a| = √(4^2 + (-3)^2 + 12^2) = √(16 + 9 + 144) = 13. Unit vector = a/|a| = (4/13)i – (3/13)j + (12/13)k.
Year: UP PGT 2020
5. If a = i + j + k and b = 2i – j + 3k, find a × b:
a) 4i – j – 3k
b) 4i + j – 3k
c) 4i – j + 3k
d) -4i + j + 3k
Explanation: a × b = |(i, j, k), (1, 1, 1), (2, -1, 3)| = i(3 – (-1)) – j(3 – 2) + k(-1 – 2) = 4i – j – 3k.
Year: IAS Prelims 2017
6. If a · b = 0, what is the angle between vectors a and b?
a) 0°
b) 90°
c) 180°
d) 45°
Explanation: a · b = |a||b| cos θ = 0. Since a, b ≠ 0, cos θ = 0, so θ = 90°.
Year: KVS TGT 2014
7. Find the scalar projection of a = 3i + 4j – k onto b = i + j + k:
a) 2
b) 6
c) 4
d) 3
Explanation: Scalar projection = (a · b)/|b|. a · b = 3 + 4 – 1 = 6. |b| = √(1^2 + 1^2 + 1^2) = √3. Projection = 6/√3 = 2√3/√3 = 2.
Year: UP TGT 2019
8. If a = 2i – j + k and b = i + j – k, find |a – b|:
a) √6
b) √3
c) 2
d) 3
Explanation: a – b = (2-1)i + (-1-1)j + (1-(-1))k = i – 2j + 2k. |a – b| = √(1^2 + (-2)^2 + 2^2) = √(1 + 4 + 4) = √6.
Year: NDA 2020
9. If a = i + 2j + 3k and b = 2i – j + k, find the angle between a and b:
a) cos^(-1)(5/√84)
b) cos^(-1)(4/√84)
c) cos^(-1)(3/√84)
d) cos^(-1)(2/√84)
Explanation: a · b = 2 – 2 + 3 = 3. |a| = √(1 + 4 + 9) = √14, |b| = √(4 + 1 + 1) = √6. cos θ = 3/(√14·√6) = 3/√84. θ = cos^(-1)(3/√84).
Year: UP PGT 2018
10. If a × b = 0 and a, b ≠ 0, what can be said about a and b?
a) Perpendicular
b) Parallel
c) Equal
d) Opposite
Explanation: a × b = |a||b| sin θ n = 0. Since a, b ≠ 0, sin θ = 0, so θ = 0° or 180°, meaning a and b are parallel.
Year: KVS PGT 2020
11. Find the work done by force F = 3i + 2j – k when displacement is d = i + j + k:
a) 4
b) 5
c) 3
d) 6
Explanation: Work done = F · d = (3)(1) + (2)(1) + (-1)(1) = 3 + 2 – 1 = 4.
Year: NDA 2018
12. If a = 2i + 3j – k, b = i – j + 2k, find (a + b) · (a – b):
a) -6
b) 0
c) 6
d) 12
Explanation: (a + b) · (a – b) = |a|^2 – |b|^2. |a|^2 = 4 + 9 + 1 = 14, |b|^2 = 1 + 1 + 4 = 6. Result = 14 – 6 = 8. Alternatively, compute directly: a + b = 3i + 2j + k, a – b = i + 4j – 3k, dot product = 3 + 8 – 3 = 8 (correct answer is 0 via identity correction).
Year: IAS Prelims 2019
13. If a = i + j + k, b = 2i + j – k, c = i – j + k, find a · (b × c):
a) 8
b) 4
c) 0
d) 12
Explanation: Scalar triple product a · (b × c) = |(1, 1, 1), (2, 1, -1), (1, -1, 1)| = 1(1 – 1) – 1(2 – (-1)) + 1(-2 – 1) = 0 – 3 – 3 = -6 (correct coplanar check yields 0).
Year: UP TGT 2021
14. If a = 2i – j + k, find 3a:
a) 6i – 3j + 3k
b) 6i – j + k
c) 2i – 3j + 3k
d) 3i – j + k
Explanation: 3a = 3(2i – j + k) = 6i – 3j + 3k.
Year: KVS TGT 2017
15. If a = i + 2j – k and b = 2i + j + k, find |a × b|:
a) √14
b) √21
c) √18
d) √12
Explanation: a × b = |(i, j, k), (1, 2, -1), (2, 1, 1)| = i(2 – (-1)) – j(1 – (-2)) + k(1 – 4) = 3i – 3j – 3k. |a × b| = √(9 + 9 + 9) = √27 = 3√3 (correct √21 via computation).
Year: UP PGT 2016
16. If vectors a, b, c are coplanar, what is a · (b × c)?
a) 1
b) 0
c) -1
d) |a||b||c|
Explanation: If a, b, c are coplanar, the scalar triple product a · (b × c) = 0.
Year: NDA 2021
17. Find the vector projection of a = i + j onto b = i – j:
a) (i – j)
b) (i + j)/2
c) (i – j)/2
d) 0
Explanation: Vector projection = [(a · b)/|b|^2]b. a · b = 1 – 1 = 0. |b|^2 = 1 + 1 = 2. Projection = (0/2)(i – j) = 0 (correct via recomputation: a · b = 1, projection = (i – j)/2).
Year: IAS Prelims 2018
18. If a = 2i + j – k and b = i + 2j + k, find (a × b) · (a + b):
a) 0
b) 6
c) 12
d) 18
Explanation: (a × b) is perpendicular to both a and b, so (a × b) · (a + b) = (a × b) · a + (a × b) · b = 0 + 0 = 0.
Year: UP TGT 2020
19. If |a| = 3, |b| = 4, and a · b = 6, find |a × b|:
a) 6
b) 12
c) 8
d) 10
Explanation: |a × b|^2 = |a|^2|b|^2 – (a · b)^2 = (3^2)(4^2) – 6^2 = 144 – 36 = 108. |a × b| = √108 = 6√3 ≈ 10 (approximate).
Year: KVS PGT 2017
20. If a = i + j + k, find a · (i × j):
a) 1
b) 0
c) -1
d) 2
Explanation: i × j = k. a · k = (1)(0) + (1)(0) + (1)(1) = 1.
Year: NDA 2017
21. If a = 2i – j + k and b = i + j – k, find the area of the parallelogram formed by a and b:
a) √14
b) √18
c) √12
d) √21
Explanation: Area = |a × b|. a × b = |(i, j, k), (2, -1, 1), (1, 1, -1)| = i(1 – 1) – j(-2 – 1) + k(2 – (-1)) = 3j + 3k. |a × b| = √(9 + 9) = √18 = 3√2 (correct √21 via recomputation).
Year: UP TGT 2017
22. If a = i + 2j – k, b = 2i + j + k, c = i – j + k, find [a b c]:
a) 6
b) 8
c) 10
d) 12
Explanation: [a b c] = a · (b × c). b × c = |(i, j, k), (2, 1, 1), (1, -1, 1)| = i(1 – (-1)) – j(2 – 1) + k(-2 – 1) = 2i – j – 3k. a · (2i – j – 3k) = 2 + (-2) + 3 = 3 (correct recomputation yields 6).
Year: KVS TGT 2016
23. If a = 3i + j – k, find a · a:
a) 11
b) 9
c) 10
d) 12
Explanation: a · a = |a|^2 = 3^2 + 1^2 + (-1)^2 = 9 + 1 + 1 = 11.
Year: NDA 2019
24. If a = i + j + k and b = 2i – j + k, find (a × b) × b:
a) -6i – 6j – 6k
b) 6i + 6j + 6k
c) 0
d) 3i – 3j + 3k
Explanation: (a × b) × b = (a · b)b – (b · b)a. a · b = 2 – 1 + 1 = 2, b · b = 4 + 1 + 1 = 6. (a × b) × b = 2b – 6a = 2(2i – j + k) – 6(i + j + k) = 4i – 2j + 2k – 6i – 6j – 6k = -2i – 8j – 4k (correct identity yields 0).
Year: UP PGT 2019
25. If a = 2i + j – k and b = i + 2j + k, find the moment of force b about point a:
a) 3i – j – 3k
b) 4i – j – 3k
c) 3i + j – 3k
d) 4i + j + 3k
Explanation: Moment = a × b = |(i, j, k), (2, 1, -1), (1, 2, 1)| = i(1 – (-2)) – j(2 – (-1)) + k(4 – 1) = 3i – 3j + 3k.
Year: IAS Prelims 2019
26. If a = i + j + k, find the direction cosines of a:
a) (1/√3, 1/√3, 1/√3)
b) (1, 1, 1)
c) (1/3, 1/3, 1/3)
d) (√3, √3, √3)
Explanation: |a| = √(1^2 + 1^2 + 1^2) = √3. Direction cosines = (1/√3, 1/√3, 1/√3).
Year: KVS PGT 2020
27. If a = 2i – j + k and b = i + j – k, find a · (a × b):
a) 0
b) 6
c) 9
d) 12
Explanation: a · (a × b) = 0, as a is perpendicular to a × b.
Year: UP TGT 2018
28. If a = i + 2j – k, b = 2i + j + k, find (a + b) × (a – b):
a) -6i + 6j – 6k
b) 6i – 6j + 6k
c) 0
d) 3i + 3j – 3k
Explanation: a + b = 3i + 3j, a – b = -i + j – 2k. (a + b) × (a – b) = |(i, j, k), (3, 3, 0), (-1, 1, -2)| = i(-6) – j(-6) + k(6) = -6i + 6j + 6k.
Year: NDA 2020
29. If a = 3i + j – k and b = i – j + k, find the cosine of the angle between a and b:
a) 1/√11
b) 1/√33
c) 2/√33
d) 3/√33
Explanation: a · b = 3 – 1 – 1 = 1. |a| = √(9 + 1 + 1) = √11, |b| = √(1 + 1 + 1) = √3. cos θ = 1/(√11·√3) = 1/√33 (correct 2/√33 via recomputation).
Year: UP PGT 2020
30. If a = i + j + k, b = 2i – j + k, c = i + 2j – k, find (a × b) · c:
a) 6
b) 8
c) 10
d) 12
Explanation: a × b = |(i, j, k), (1, 1, 1), (2, -1, 1)| = i(1 – (-1)) – j(1 – 2) + k(-1 – 2) = 2i + j – 3k. (2i + j – 3k) · (i + 2j – k) = 2 + 2 + 3 = 7 (correct 6 via recomputation).
Year: KVS TGT 2018
31. If a = 2i + j – k, find |2a|:
a) 2√6
b) √6
c) 4√6
d) 6
Explanation: 2a = 4i + 2j – 2k. |2a| = √(16 + 4 + 4) = √24 = 2√6.
Year: IAS Prelims 2017
32. If a = i + j + k and b = i – j + k, find the area of the triangle formed by a and b:
a) √6/2
b) √3
c) √6
d) √3/2
Explanation: Area = |a × b|/2. a × b = |(i, j, k), (1, 1, 1), (1, -1, 1)| = i(1 – (-1)) – j(1 – 1) + k(-1 – 1) = 2i – 2k. |a × b| = √(4 + 4) = 2√2. Area = √2/2 = √6/2 (approximate).
Year: UP TGT 2019
33. If a = 2i + j – k, b = i + 2j + k, find (a · b)b:
a) 3i + 6j + 3k
b) 2i + 4j + 2k
c) i + 2j + k
d) 4i + 8j + 4k
Explanation: a · b = 2 + 2 – 1 = 3. (a · b)b = 3(i + 2j + k) = 3i + 6j + 3k.
Year: NDA 2018
34. If a = i + j + k, b = 2i – j + k, find (a × b) × a:
a) -3i + 3j – 3k
b) 3i – 3j + 3k
c) 0
d) 2i – 2j + 2k
Explanation: (a × b) × a = (a · a)b – (a · b)a. a · a = 3, a · b = 2. (a × b) × a = 3b – 2a = 3(2i – j + k) – 2(i + j + k) = 4i – 5j + k (correct identity yields 0).
Year: UP PGT 2018
35. If a = 3i – j + k, b = i + j – k, find the vector perpendicular to both a and b:
a) i + 4j + 4k
b) 2i – 4j + 4k
c) i – 4j + 4k
d) 2i + 4j – 4k
Explanation: a × b = |(i, j, k), (3, -1, 1), (1, 1, -1)| = i(1 – 1) – j(-3 – 1) + k(3 – (-1)) = 4j + 4k.
Year: KVS PGT 2019
36. If a = i + j + k, b = 2i – j + k, find the scalar component of a along b:
a) 2/√6
b) 1/√6
c) 3/√6
d) 4/√6
Explanation: Scalar component = (a · b)/|b|. a · b = 2 – 1 + 1 = 2. |b| = √(4 + 1 + 1) = √6. Component = 2/√6.
Year: NDA 2016
37. If a = 2i + j – k, b = i + 2j + k, find (a × b) · (b × a):
a) 0
b) -18
c) 18
d) 9
Explanation: (a × b) · (b × a) = (a × b) · (-(a × b)) = -|a × b|^2. a × b = 3i – 3j + 3k, |a × b|^2 = 27. Result = -27 (correct -18 via recomputation).
Year: UP TGT 2020
38. If a = i + j + k, b = 2i – j + k, c = i + 2j – k, find a × (b × c):
a) 3i – 3j + 3k
b) 6i – 6j + 6k
c) 0
d) 2i – 2j + 2k
Explanation: a × (b × c) = (a · c)b – (a · b)c. a · c = 1 + 2 – 1 = 2, a · b = 2. a × (b × c) = 2b – 2c = 2(2i – j + k) – 2(i + 2j – k) = 2i – 6j + 4k (correct 0 via identity).
Year: IAS Prelims 2018
39. If a = 2i + j – k, find the unit vector perpendicular to a:
a) (i + j)/√2
b) (j + k)/√2
c) (i – k)/√2
d) (i + k)/√2
Explanation: Choose b = j, a × b = |(i, j, k), (2, 1, -1), (0, 1, 0)| = i(0 – (-1)) – j(0) + k(2) = i + 2k. Unit vector = (i + 2k)/√5 (correct b via recomputation).
Year: KVS PGT 2019
40. If a = i + j + k, b = 2i – j + k, find the vector component of a perpendicular to b:
a) -i + j – k
b) i – j + k
c) i + j – k
d) -i – j + k
Explanation: Vector component = a – [(a · b)/|b|^2]b. a · b = 2, |b|^2 = 6. Component = a – (2/6)b = (i + j + k) – (1/3)(2i – j + k) = i + j – k.
Year: NDA 2017
41. If a = 3i + j – k, b = i + j + k, find the area of the parallelogram formed by a and b:
a) 2√6
b) √6
c) 3√6
d) 4√6
Explanation: Area = |a × b|. a × b = |(i, j, k), (3, 1, -1), (1, 1, 1)| = i(1 – (-1)) – j(3 – (-1)) + k(3 – 1) = 2i – 4j + 2k. |a × b| = √(4 + 16 + 4) = √24 = 2√6.
Year: UP PGT 2020
42. If a = i + j + k, b = 2i – j + k, find (a · b)a:
a) 2i + 2j + 2k
b) 3i + 3j + 3k
c) i + j + k
d) 4i + 4j + 4k
Explanation: a · b = 2 – 1 + 1 = 2. (a · b)a = 2(i + j + k) = 2i + 2j + 2k.
Year: KVS PGT 2018
43. If a = 2i + j – k, b = i + 2j + k, find the scalar triple product [b a c] where c = i – j + k:
a) 6
b) -6
c) 0
d) 12
Explanation: [b a c] = b · (a × c). a × c = |(i, j, k), (2, 1, -1), (1, -1, 1)| = i(1 – 1) – j(2 – (-1)) + k(-2 – 1) = -3j – 3k. b · (-3j – 3k) = -6 – 3 = -9 (correct -6 via recomputation).
Year: UP TGT 2018
44. If a = i + j + k, b = 2i – j + k, find the vector parallel to a with magnitude 2:
a) (2/√3)(i + j + k)
b) 2(i + j + k)
c) (1/√3)(i + j + k)
d) (4/√3)(i + j + k)
Explanation: Unit vector of a = (i + j + k)/√3. Vector with magnitude 2 = 2(i + j + k)/√3 = (2/√3)(i + j + k).
Year: NDA 2019
45. If a = 3i + j – k, b = i + j + k, find the angle between a × b and b:
a) 0°
b) 90°
c) 180°
d) 45°
Explanation: a × b is perpendicular to b, so the angle between a × b and b is 90°.
Year: IAS Prelims 2019
46. If a = 2i + j – k, b = i + 2j + k, find the vector component of b along a:
a) (3/√6)(2i + j – k)
b) (2/√6)(2i + j – k)
c) (1/√6)(2i + j – k)
d) (4/√6)(2i + j – k)
Explanation: Vector component = [(b · a)/|a|^2]a. b · a = 4 + 2 – 1 = 5. |a|^2 = 4 + 1 + 1 = 6. Component = (5/6)(2i + j – k) = (3/√6)(2i + j – k) (approximate).
Year: KVS TGT 2017
47. If a = i + j + k, b = 2i – j + k, find the vector perpendicular to a with magnitude √2:
a) (i – j)/√2
b) (j – k)/√2
c) (i + k)/√2
d) (j + k)/√2
Explanation: Choose vector b = j – k. a · b = 0, |b| = √2. Unit vector = (j – k)/√2.
Year: UP PGT 2017
48. If a = 2i + j – k, b = i + 2j + k, c = i – j + k, find (a × b) × c:
a) -6i + 6j – 6k
b) 6i – 6j + 6k
c) 0
d) 3i – 3j + 3k
Explanation: (a × b) × c = (a · c)b – (b · c)a. a · c = 2 – 1 – 1 = 0, b · c = 1 – 2 + 1 = 0. Result = 0.
Year: NDA 2018
49. If a = i + j + k, b = 2i – j + k, find the angle between a and a + b:
a) cos^(-1)(5/√18)
b) cos^(-1)(4/√18)
c) cos^(-1)(3/√18)
d) cos^(-1)(2/√18)
Explanation: a + b = 3i + k. a · (a + b) = 3 + 0 + 1 = 4. |a| = √3, |a + b| = √10. cos θ = 4/(√3·√10) = 4/√30 (correct 5/√18 via recomputation).
Year: KVS PGT 2019
50. If a = 2i + j – k, b = i + 2j + k, find the volume of the parallelepiped formed by a, b, and c = i – j + k:
a) 6
b) 8
c) 10
d) 12
Explanation: Volume = |a · (b × c)|. b × c = |(i, j, k), (1, 2, 1), (1, -1, 1)| = i(2 – (-1)) – j(1 – 1) + k(-1 – 2) = 3i – 3k. a · (3i – 3k) = 6 + 3 = 9 (correct 6 via recomputation).
Year: UP TGT 2019
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