Answer 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 cosθ `hatn`

= abc cos 0°

= abc

= Volume of the parallelepiped

Answer 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