Maths · Vectors and 3-D geometry · chapter 10 of 13

Class 12 Maths Chapter 10: Vector Algebra MCQs with answers

10 multiple-choice questions on Vector Algebra, each with its answer and the reason, written from the NCERT chapter. For the CBSE board and JEE Main, which are set from the same NCERT chapter. Open a question's solution to check your working; then take the free test to find your next chapter.

Question 1
Which one of the following is a vector quantity?
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Answer: (b) the weight, 50 N, of a sack of rice

Weight is a force: it has a magnitude and a direction (vertically down). Energy, speed and temperature have magnitude only, so they are scalars.

Question 2
The magnitude of the vector 4î − 4ĵ + 7k̂ is
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Answer: (c) 9

|4î − 4ĵ + 7k̂| = √(16 + 16 + 49) = √81 = 9. Adding the components gives 7; dropping one square gives √65; 81 is the square of the magnitude.

Question 3
The vector 12î + yĵ − 4k̂, with y > 0, has magnitude 13. The value of y is
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Answer: (d) 3

144 + y2 + 16 = 169, so y2 = 9 and, as y > 0, y = 3. Stopping at y2 gives 9.

Question 4
A vector has direction cosines l = 27, m = −37 and n, where n < 0. Then n is
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Answer: (a) −67

l2 + m2 + n2 = 1 gives n2 = 1 − 449 − 949 = 3649, so n = ±67; with n < 0, n = −67. 67 ignores the condition n < 0.

Question 5
For P(3, −1, 4) and Q(−2, 5, 1), the scalar component of →QP along ĵ is
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Answer: (b) −6

→QP = →OP − →OQ = (3 + 2)î + (−1 − 5)ĵ + (4 − 1)k̂ = 5î − 6ĵ + 3k̂, so the ĵ-component is −6. 6 is the component of →PQ; 4 comes from adding the ĵ-coordinates.

Question 6
If (p − 2)î + 3ĵ − qk̂ = 5î + (r + 1)ĵ + 4k̂, then p + q + r equals
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Answer: (c) 5

Equal vectors have equal components: p − 2 = 5, 3 = r + 1, −q = 4. So p = 7, r = 2, q = −4 and p + q + r = 5. Taking q = 4 gives 13.

Question 7
If →a = 3î − 2ĵ + k̂ and →b = 2î + 4ĵ − k̂, then →a · →b is
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Answer: (d) −3

→a · →b = a₁b₁ + a₂b₂ + a₃b₃ = 6 − 8 − 1 = −3. Taking the k̂ term as +1 gives −1; taking (−2)(4) as +8 gives 13.

Question 8
The vectors 2î + λĵ − 3k̂ and 4î − ĵ + 2k̂ are perpendicular when λ equals
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Answer: (a) 2

Perpendicular means the scalar product is 0: 8 − λ − 6 = 0, so λ = 2. Adding −6 as +6 gives 14.

Question 9
→b is opposite in direction to →a = 2î − 3ĵ + 6k̂ and |→b| = 14. Then →a · →b equals
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Answer: (b) −98

|→a| = √(4 + 9 + 36) = 7. Opposite directions mean θ = π, so →a · →b = |→a||→b| cos π = −7 × 14 = −98. Forgetting cos π = −1 gives 98; 0 would need perpendicular vectors.

Question 10
The vector k̂ × î − ĵ × k̂ equals
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Answer: (c) ĵ − î

k̂ × î = ĵ and ĵ × k̂ = î, so k̂ × î − ĵ × k̂ = ĵ − î. Reversing either product changes a sign.

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