Show that $\text{a}.(\text{b}\times \text{c})$ is equal in magnitude to the volume of the parallelopiped formed on the three vectors, $\text{a, b}$ and $\text{c}$.

A parallelopiped with origin O and sides $a,b,$ and $c$ is shown in the following figure.

Volume of parallelopiped=$abc$

Let $\hat{n}$ be a unit vector perpendicular to both $\vec{b}$ and $\vec{c}$. Hence $\hat{n}$ and $\vec{c}$ have the same direction.

Therefore, $\vec{b}\times \vec{c}=bcsin\theta \hat{n}$$=bc\hat{n}$

$\vec{a}.(\vec{b}\times \vec{c})=a.(bc\hat{n})$

                  $=abc=$ Volume of parallelopiped