Question

Show that a.(b × c) is equal in magnitude to the volume of the parallelepiped formed on the three vectors , a, b and c.

Answer 1

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

Volume of the given parallelepiped = abc
$\vec {OC} = \vec a$
$\vec {OB} = \vec b$
$\vec {OC} = \vec c$

Let be a unit vector perpendicular to both b and c. Hence, n^ and a have the same direction.

∴ $\vec b$ × $\vec c$ = bc sinθ $\^n$

= bc sin 90° $\^n$

= bc$\^n$

$\vec a.(\vec b × \vec c)$

=$a.(bc\^n)$

= abc cosθ $\^n$

= abc cos 0°

= abc

= Volume of the parallelepiped