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*(b* c)=(a\cdot c)b-(a\cdot b)c`

So

`(v* w)*(w* u)=((v* w)\cdot u)w-((v* w)\cdot w)u`

Also recall the scalar triple product,

`a\cdot(b* 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* w)\cdot w=0`

and the above reduces to

`(v* w)*(w* u)=((v* w)\cdot u)w`

so that

`(u* v)\cdot[(v* w)*(w* u)]=(u* v)\cdot((v* w)\cdot u)w`

The scalar triple product has the following property:

`a\cdot(b* c)=b\cdot(c* a)=c\cdot(a* b)`

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

`((v* w)\cdot u)((u* v)\cdot w)`

and by the property above we have

`(u* v)\cdot w=u\cdot(v* w)`

and so we end up with

`[u\cdot(v* w)]^2`

as required.