Vector Algebra – important questions for Boards 2027
CBSE Class 12 Maths
Find |3î + 4ĵ|.
Answer
|3î + 4ĵ| = √(3² + 4²) = √25 ∴ |3î + 4ĵ| = 5
Which statement correctly describes the zero vector? (A) It has magnitude 1 and a definite direction. (B) Its initial and terminal points coincide, and it has no definite direction. (C) Its initial and terminal points are distinct, and it has zero magnitude. (D) It has the same magnitude as every unit vector.
Answer
(B) Its initial and terminal points coincide, and it has no definite direction — a zero vector has coincident initial and terminal points and zero magnitude.
Determine whether a = 2î + 3ĵ − k̂ and b = 2î − 3ĵ − k̂ are equal by comparing their corresponding components.
Answer
a = (2, 3, −1) b = (2, −3, −1) 3 ≠ −3 ∴ a ≠ b
For which value of t are the vectors 2î + 3ĵ − k̂ and 2î + tĵ − k̂ equal? (A) −1 (B) 0 (C) 2 (D) 3
Answer
(D) 3 — equal vectors have equal corresponding components, so t = 3.
Find the value of λ for which a⃗ = λî + 2ĵ − k̂ and b⃗ = 2î − ĵ + 4k̂ are perpendicular. Verify your value using their component-form scalar product.
Answer
Since a and b are perpendicular, a · b = 0. (λî + 2ĵ − k̂) · (2î − ĵ + 4k̂) = 0 2λ − 2 − 4 = 0 2λ = 6 ∴ λ = 3 For λ = 3, a = 3î + 2ĵ − k̂. a · b = 3(2) + 2(−1) + (−1)(4) = 6 − 2 − 4 ∴ a · b = 0, verifying that a and b are perpendicular.
Find the projection vector of a⃗=2î+3ĵ+2k̂ on b⃗=2î+ĵ+k̂, and hence find the magnitude of the projection of a⃗ on b⃗.
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For vectors a⃗ = a₁î + a₂ĵ + a₃k̂ and b⃗ = b₁î + b₂ĵ + b₃k̂, describe how multiplication of a⃗ by a scalar affects its components. State the distributive laws for vector addition and scalar multiplication. Also state the condition for two vectors to be collinear, including its component-form equivalent. (5 marks)
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Find the scalar projection of 2î on î.
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A nonzero vector a has magnitude 5. Which expression gives the unit vector in its direction? (A) 5a (B) a/5 (C) a + 5 (D) 5/a
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Find the vector directed from P(1, −2, 3) to Q(4, 2, −1).
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72 more probable questions in this chapter, every one with a full answer.
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