Question 7. 5. Show that a.b x c is equal in magnitude to the volume of the parallelepiped formed on the three vectors, a, b and c.


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

Volume of the given parallelepiped = abc
OC=a
OB=b
OC=c
Let n^ be a unit vector perpendicular to both b and c. Hence, n^ and a have the same
direction.
b×c=bcsinθ n^
=bc sin 90° n^
=bc n^
a·b×c
=a·(bcn^)
=abc cosθ n^
=abc cos 0°
=abc
=Volume of the parallelpiped