Recall that for 3 vectors
, all in
, the vector triple product

So

Also recall the scalar triple product,

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

and the above reduces to

so that
![(u* v)\cdot[(v* w)*(w* u)]=(u* v)\cdot((v* w)\cdot u)w](https://img.qammunity.org/2020/formulas/mathematics/college/dul85idrfl90y9vl788fk3w5lz54qnxqfy.png)
The scalar triple product has the following property:

Since
is a scalar, we can factor it out to get

and by the property above we have

and so we end up with
![[u\cdot(v* w)]^2](https://img.qammunity.org/2020/formulas/mathematics/college/bnoyk85linq7nd25gu62mfi37ptcs88leq.png)
as required.