Question

Let a,b and c be three unit vectors such that a.b = a.c = 0 and the angle between b and c is π/6, prove that a = ±2(b × c).

Answer 1

Since a.b = a.c = 0

∴ a is ⊥ to the plane b and c

Also, \(b \times c = |b| |c|\sin\frac \pi 6 \hat n = \frac 12 \hat n\)

But a = \( \hat n\), both being unit vectors ⊥ to the plane of b and c

a = ±2(b × c)

Hence proved.

Answer 2

Given

a.b = 0 = a.c

Thus, a is perpendicular to b and c.

A unit vector perpendicular to b and c 

= ±2(b × c).