Orthogonal Vectors and Projections

Question

Given that u and v are orthogonal unit vectors in $R^3$, prove that for all x, where x is a vector in R3 as well, that:

$perp_{u×v}(x)=proj_u(x)+proj_v(x)$

So far, I've tried using the definition of both projection and perpendicular but it seems too complicated to try and manipulate them. Any help would be greatly appreciated.

Answer 1

Hint: Recall that the cross product satisfies $(u \times v) \cdot u = 0$ and $(u \times v) \cdot v = 0$. Since $u, v$ are orthogonal unit vectors, $||u \times v|| = ||u||||v|| = 1$, and so $u, v, u \times v$ are mutually orthogonal unit vectors.

Additional hint: Recall that for all vectors $a, b$ we have $$b = \text{proj}_a(b) + \text{perp}_a(b),$$ so we can write both sides just terms of $\text{proj}_{\ast}(\,\cdot\,)$.