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Cross Product

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Question

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

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Solution

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

Volume of the given parallelepiped = abc
$¯ O C = \to a ¯ O B = \to b ¯ O C = \to c$
Let $^n$ be a unit vector perpendicular to both b and c. Hence, $^n$ and $\to a$ have the same direction.
$∴ \to b \times \to c = b c s i n θ^n = b c s i n 90^{\circ}^n = b c^n \to a . (\to b \times \to c) = \to a . (b c^n) = a b c c o s θ^n = a b c c o s 0^{\circ} = a b c$
= Volume of the parallelepiped

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