Proving a cross product satisfies the vector equation

Question

Let vectors $u,v,w \in R^3$

Prove that $u \times (v \times w)$ must be a vector that satisfies the vector equation $x=sv+tw$ where $s,t \in R$

I have no idea where to go with this one, any tips?

Answer 1

Use the identity $$u \times (v \times w)=(u \cdot w)v-(u \cdot v) w$$ The proof is there and more obvious there

Answer 2

If $v,w$ are linearly dependent, this product vanishes and it is obviously a linear combination, as desired. If $v,w$ are independent, then $v\times w$ is orthogonal to the plane spanned by $v,w$, and the vector $x=u\times(v\times w)$ is orthogonal to $v\times w$, and thus lies in $span(v,w)$.