Proof in a Scalar Triple Product

Question

For any three vectors $\vec a,\ \vec b,\ \vec c$, show that :

$$[\vec a\times\vec b,\ \vec b\times\vec c,\ \vec c\times\vec a]=[\vec a,\ \vec b,\ \vec c]^2$$

where $[\vec a,\vec b,\vec c]=\vec a\cdot (\vec b\times \vec c)$. Does it employ properties of vector triple product? Please explain the properties involved. We were not taught vector triple product at school. Thanks

Answer 1

$$\begin{align}&[a\times b,b\times c,c\times a]\\ =&(a\times b)\cdot((b\times c)\times(c\times a))\\ =&(a\times b)\cdot(((b\times c)\cdot a)c-((b\times c)\cdot c)a)\\ =&((a\times b)\cdot c)((b\times c)\cdot a)\\ =&[a,b,c]^2\end{align}$$

The first equality follows directly from the definition. The second equality uses the fact $a\times(b\times c)=(a\cdot c)b-(a\cdot b)c$. And the third results from $b\times c$ is perpendicular to $c$. The last uses the symmetry of triple scalar product.

Answer 2

As everything's linear, I'd try to prove it for $a = e_x, b = e_y, c = e_z$ and then go to proving the statement for linear combinations of $e_x, e_y, e_z$.