Q

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

Q7.5     Show that $a.(b\times c)$ is equal in magnitude to the volume of the parallelepiped formed
on the three vectors , $a$$b$ and $c$ .

Views

A parallelepiped is shown in the figure given below:-

Volume is given by :    =  abc

We can write :

$|b\times c|\ =\ |b||c|\sin \Theta\ \widehat{n}$                           (The direction of $\widehat{n}$ is in the direction of vector a.)

$=\ |b||c|\sin 90^{\circ}\ \widehat{n}$

$=\ |b||c|\ \widehat{n}$

Now,

$a.(b\times c)\ =\ a.(bc)\ \widehat{n}$

$=\ abc \cos \Theta$

$=\ abc \cos 0^{\circ}$

$=\ abc$

This is equal to volume of parallelepiped.

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