Vector Triple Product:
The
vector product of two vectors, one of which is itself the vector product
of two vectors, is a vector quantity called vector triple product.
It
is defined for three vectors
as the vector
. This vector being perpendicular to
, is coplanar with
.
i.e.

Take
the scalar product of this equation with a. We get.
0
= 
&⇒

If
we choose the coordinate axes in such a way that
, it is easy to show that l = 1. Hence

In
general,
( Vector triple product is not associative
) .
if some or all of
are
zero vectors or
are collinear.
Example.1
Prove that 
Solution
We have
= 
=
=
.
Example.2
Show that
= 0.
Solution
L.H.S. = 

=0
Example.3
Prove that
.
Solution
L.H.S. =
= 
=
= R.H.S.
Example.4
If are
three unit vectors such that = , find the angles which makes with and
..
Solution
We are given that =
&⇒ 
&⇒
and
= 0
Hence the angle between
is
and that between,
.