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.

Answer 1

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 `vec(OA)` 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.