Question

Let: u, v, w \varepsilon R3 Prove: (u x v) * [(v x w) x (w x u)] = [u * (v x w)]2

Answer 1

Recall that for 3 vectors $a,b,c$, all in $\mathbb R^3$, the vector triple product

$a\times(b\times c)=(a\cdot c)b-(a\cdot b)c$

So

$(v\times w)\times(w\times u)=((v\times w)\cdot u)w-((v\times w)\cdot w)u$

Also recall the scalar triple product,

$a\cdot(b\times c)$

which gives the signed volume of the parallelipiped generated by the three vectors $a,b,c$. When either $a=b$ or $a=c$, the parallelipepid is degenerate and has 0 volume, so

$(v\times w)\cdot w=0$

and the above reduces to

$(v\times w)\times(w\times u)=((v\times w)\cdot u)w$

so that

$(u\times v)\cdot[(v\times w)\times(w\times u)]=(u\times v)\cdot((v\times w)\cdot u)w$

The scalar triple product has the following property:

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

Since $(v\times w)\cdot u$ is a scalar, we can factor it out to get

$((v\times w)\cdot u)((u\times v)\cdot w)$

and by the property above we have

$(u\times v)\cdot w=u\cdot(v\times w)$

and so we end up with

$[u\cdot(v\times w)]^2$

as required.