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