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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 , ab and c .

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A parallelepiped is shown in the figure given below:- 

                                            Rotational motion ,   20116

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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