Question

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 $\text{c}$.

Answer 1

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 $\overrightarrow{b}$ and $\overrightarrow{c}$. Hence $\hat{n}$ and $\overrightarrow{c}$ have the same direction.

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

$\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c})=a.\left(bc\hat{n}\right)$

$=abc=$ Volume of parallelopiped