If a⃗ and b⃗ are perpendicular, then
a⃗ × ( a⃗ × ( a⃗ × ( a⃗ × b⃗ ) ) ) is equal to :

Solution

Maths sir · 3 steps

Expand the innermost two crosses once, and the same pattern repeats for the outer two.

1. Triple product rule. Use a⃗ × (b⃗ × c⃗) = (a⃗ · c⃗) b⃗ - (a⃗ · b⃗) c⃗. Since a⃗ ⊥ b⃗, a⃗ · b⃗ = 0.

   a⃗ × (a⃗ × b⃗) = (a⃗ · b⃗) a⃗ - (a⃗ · a⃗) b⃗ = -|a⃗|² b⃗

2. Repeat outside. The given expression is a⃗ × (a⃗ × (-|a⃗|²b⃗)) = -|a⃗|² a⃗ × (a⃗ × b⃗). Use the result above once more: -|a⃗|² × (-|a⃗|² b⃗).
3. So. The two minus signs cancel and the expression equals |a⃗|⁴ b⃗, which is option (d).

The idea to keep. When a⃗ ⊥ b⃗, crossing with a⃗ twice turns b⃗ into -|a⃗|²b⃗. Every pair of crosses adds a factor of -|a⃗|².

Answer from the NTA final answer key. Solution by PrepWiser.