Question 7.5:
Show
that a.
(b
× c)
is equal in magnitude to the volume of the parallelepiped formed on
the three vectors, a,
b
and c.
Solution:
A parallelepiped with origin O and sides a, b, and c is shown in the following figure.
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 X c = bc Sin θ n^
= bc n^
a.(b X c) = a(bcn^)
= abc Cosθ n^
= abc Cos 0
= abc
= Volume of the parallelepiped
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 X c = bc Sin θ n^
= bc n^
a.(b X c) = a(bcn^)
= abc Cosθ n^
= abc Cos 0
= abc
= Volume of the parallelepiped
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