3.10*triple products*
The inneer product
is a scalar, and we can’t use the result in
further vector or dot products. The outer product
is a vector so it may be combined with a third vector
to form either a scalar product
, or a vector product:
.
We shall look at the scalar triple
product,
It is clearly a scalar quantity since
is a number. It is particularly relevant to study the
geometric interpretation, as in Fig.
3.16
.
The quantity
is the height of the parallelopiped in that
figure, adn we find that
where
is the volume of the parallelopiped.
is independent of the way it is calculated, i.e., any
face may be used as base. Hence
Since scalar product is commutative
All the six expressions are equal! The
and the
may be interchanged as long as product is
defined.
3.10.1 Component Form
We know that
then
with
,
This can be put in determinant form,
Note that the order of the columns rows is the
same as the order of the vectors.
,
and
in the STP.
Example
3.12:
-
Find
given
,
,
.
3.10.2 Some physical examples
Important physical quantities represented
by a vector product are
- Angular momentum: This is defined
as the product of the distance from a centre with the
momentum perpendicular to this distance;
- Magnetic force. The force on a charged
particle (charge
) moving with velocity
in a constant magnetic field
is perpindicular to both
and
, with size commensurate with the outer product
- Torque: The torque of a force describes
the rotational effect of such a force (think about
moving a crank). Clearly only the force perpendicular to the
crank makes it rotate, hence the definition