#### Question

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 1

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

Volume of the given parallelepiped = *abc*

`vecOC = veca`

`vecOB = vecb`

`vecOC = vecc`

Let `hatn` be a unit vector perpendicular to both *b* and *c*. Hence, `hatn` and *a* have the same direction.

`:. vecb xx vecc = bc sin theta hatn`

`=bc sin 90^@ hatn`

`=bc sin 90^@ hatn`

`= bc hatn`

`veca.(vecb xx vecc)`

`= a.(bc hatn)`

= *abc *cos*θ* `hatn`

= *abc* cos 0°

= *abc*

= Volume of the parallelepiped

#### Solution 2

Let a parallelopiped be formed on the three vectors.

`vec(OA) = veca`, `vec(OB) = vecb` and `vec(OC) = vecC`

Now `vecb xx vecc = bc sin 90^@ hatn = bc hatn`

where `hatn` is unit vector along `vecOA` perpendicular to the plane containing `vecb` and `vecc`

Now `veca.(vecb xx vecc) = veca.bc hatn`

`=(a)(bc) cos 0^@`

= abc

which is equal in magnitude to the volume of the parallelopiped