JEE Main 2021 · 18 March · evening · Single correct · +4 / −1
Let a⃗ and b⃗ be two non-zero vectors perpendicular to each other and |a⃗ | = |b⃗ |. If |a⃗ × b⃗ | = |a⃗ |, then the angle between the vectors ( a⃗ + b⃗ + ( a⃗ × b⃗ ) ) and a⃗ is equal to :
- (a) sin ⁻¹( 1/√6 )
- (b) cos ⁻¹( 1/√2 )
- (c) sin ⁻¹( 1/√3 )
- (d) cos ⁻¹( 1/√3 )
Answer and solution
Answer (d) cos ⁻¹( 1/√3 )
Solution
Maths sir · 4 steps
First show both vectors are unit vectors. Then a, b and a × b act like i, j and k.
- Find the lengths. Since a⃗ ⊥ b⃗, |a⃗ × b⃗| = |a⃗||b⃗|sin 90° = |a⃗|². Setting this equal to |a⃗| gives |a⃗| = 1, and so |b⃗| = 1 too. Both vectors are non-zero, so |a⃗| = 0 is ruled out.
- Three perpendicular unit vectors. a⃗ × b⃗ is perpendicular to both a⃗ and b⃗, and its length is 1. So a⃗, b⃗ and a⃗ × b⃗ behave exactly like î, ĵ and k̂.
- Dot product and length. Let v⃗ = a⃗ + b⃗ + a⃗ × b⃗. Then v⃗ · a⃗ = 1 + 0 + 0 = 1, and |v⃗| = √(1 + 1 + 1) = √3.
cos θ = (v⃗ · a⃗)/(|v⃗| |a⃗|) = 1/(√3 × 1) = 1/√3
- So. The angle is θ = cos⁻¹(1/√3), which is option (d).
The idea to keep. Two perpendicular unit vectors and their cross product form a ready-made î, ĵ, k̂ set. Once you see that, any length or angle is just like working in coordinates.
Answer from the NTA final answer key. Solution by PrepWiser.