find $u$, $v$, $w$ such that $u \times (v \times w)\neq (u \times v) \times w$

Question

Can someone help with this question?

Let $\times$ denote the cross product of vectors in $\mathbb{R}^3$.

Find $u$, $v$, $w$ such that $u \times (v \times w)$ is different from $(u \times v) \times w$?

I don't know how to start, Thanks!

Answer 1

Let $i,j,$ and $k$ be the standard basis vectors in $\mathbb{R}^3$. Then $$i\times (i\times j) = i\times k = -j$$ $$(i\times i)\times j = 0\times j = 0$$

Answer 2

Jacob Identity

$$a\times (b\times c)+b\times(c \times a) +c\times(a \times b)=0$$

$$a\times(b\times c)=-c\times (a \times b) $$ iff$$b\times(c\times a)=0$$

Now note $-c\times (a \times b)=(a \times b)\times c$.

then $$a\times(b\times c)=(a \times b) \times c$$

So we mustn't have $$b\times (c \times a) =0$$

Or as you wrote all vectors which do not satisfy $v\times (w \times u)=0$.