Vector cross products proof

Question

Explain why $U \times ( V \times W )$ must be a vector that satisfies the equation $X = sV + tW$, where $U$, $V$, and $W$ are vectors in $\mathbb R^3$.

Answer 1

If $v \times w=0$, the conclusion is immediate with $s=t=0$.

Otherwise, $v$ and $w$ are linearly independent, which is to say they are contained in a unique plane through the origin. The vector $v \times w$ is normal to both $v$ and $w$. The vector $u \times (v \times w)$, in turn, is normal to $v \times w$ (and to $u$). So, $u \times (v\times w)$ is normal to the normal vector of the plane containing $v$ and $w$.

So, $u \times (v\times w)$ is in the plane containing $v$ and $w$, which is to say $u \times (v\times w) = sv + tw$ for some $s,t \in \Bbb R$.