JEE AdvancedHardQuestion 260014
Question
If a⃗ × b⃗ = c⃗ and b⃗ × c⃗ = a⃗, then:
- A|a⃗| = |b⃗| = |c⃗| = 1 and they are mutually perpendicularCorrect
- B|b⃗| = 1 only
- Ca⃗ = b⃗ = c⃗
- DThe vectors are coplanar
Correct answer
|a⃗| = |b⃗| = |c⃗| = 1 and they are mutually perpendicular
Explanation
From a⃗ × b⃗ = c⃗: c⃗ ⊥ a⃗ and c⃗ ⊥ b⃗. From b⃗ × c⃗ = a⃗: a⃗ ⊥ b⃗ and a⃗ ⊥ c⃗. So all three are mutually perpendicular. Also |c⃗| = |a⃗||b⃗|sin(90°) = |a⃗||b⃗|. Similarly |a⃗| = |b⃗||c⃗|. This gives |a⃗| = |b⃗||c⃗| = |b⃗|·|a⃗||b⃗| = |a⃗||b⃗|² → |b⃗|² = 1 → |b⃗| = 1. Similarly |a⃗| = |c⃗| = 1.