Cross Product Calculation Question Given Orthogonality Conditions

Question

Suppose that vectors $u$, $v$, $w$ are mutually orthogonal. Compute $(u \times v) \times w$ and $u \times (v \times w)$

Wouldn't we simply multiply $u$ and $v$ and $w$ all together? Or is there a formula I'm missing here?

Answer 1

I assume you are working in $\mathbb{R}^3$ in order for this to make sense. Now, $(u\times v)\perp u$ and $(u\times v)\perp v$, so that $u\times v$ must be a scalar multiple of $w$, i.e. $u\times v=\lambda w$. So, $(u\times v)\times w=\lambda w\times w=0.$

The computation of $u\times(v\times w)$ is analogous.