I know that this is a property of coplanar vectors but I don't know how to prove it and can't find any answers anywhere. The question is worth $6$ marks in a past paper so it will be more than a couple of lines of work:
Show that the vectors $a,b,c$ are coplanar if and only if $a\cdot (b\times c) =0$? Any help would be greatly appreciated.
On
If vectors are coplanar, $\vec b\times\vec c\perp\vec{a}\implies[\vec a,\vec b,\vec c]=\vec{0}$
Also $[\vec a,\vec b,\vec c]=\vec 0\implies\vec{b}\times{\vec c}=\vec 0$.
Last case implies they are parallel and thus coplanar.
Note that I have avoided trivial case of either of them being $\vec 0$
Hint. $a \cdot (b \times c)$ is the determinant of a matrix with rows $a, b, c$.