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

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Solution

A parallelopiped with origin O and sides $a, b,$ and $c$ is shown in the following figure.
Volume of parallelopiped= $a b c$
Let $^n$ be a unit vector perpendicular to both $\to b$ and $\to c$ . Hence $^n$ and $\to c$ have the same direction.
Therefore, $\to b \times \to c = b c s i n θ^n$ $= b c^n$
$\to a . (\to b \times \to c) = a . (b c^n)$
$= a b c =$ Volume of parallelopiped

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Show that a.(b×c) is equal in magnitude to the volume of the parallelopiped formed on the three vectors, a, b and c.

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