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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
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.
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