Question

Show that a . (b X C) is equal in magnitude to the volume of the parallelepiped formed on the three vectors a, b and c.

Answer 1

Hii dear,

● Proof-
[Refer to the figure.]

Consider a paralleopiped with origin O and sides a, b and c.

Volume of the parallelepiped
V = |a||b||c|

Consider, n̂ be unit vector perpendicular to b and c.
Thus n̂ and a will have same direction.
b × c = |b||c|sinθ.n̂
b × c = |b||c|n̂

Let's assume,
a.(b×c) = |a|.(|b||c|n̂)
a.(b×c) = |a||b||c|cos0.n̂
a.(b×c) = |a||b||c|
a.(b×c) = V
Hence, proved.

Hope that helps you...