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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 ˆn be a unit vector perpendicular to both b and c. Hence, ˆn and a have the same
direction.
b×c=bcsinθ ˆn
=bc sin 90° ˆn
=bc ˆn
a·(b×c)
=a·(bcˆn)
=abc cosθ ˆn
=abc cos 0°
=abc
=Volume of the parallelpiped