Friday, 8 August 2014
CHAPTER 24 -Scalar Triple Product
As the name suggests, a scalar triple product involves the (scalar) product of three vectors. How may such a product be defined?
Consider three vectors
. Consider the quantity
is a scalar, you cannot define its dot product with another vector. Thus,
is a meaningless quantity.
However, consider the expression
is a vector, its dot product with
is defined. Thus,
is defined and is termed the scalar triple product of
This product is represented concisely as
An alert reader might have noticed that another valid triple product is possible:
. This is the vector triple product and is considered in the next section.
Let us try to assign a geometrical interpretation to the scalar triple product
First of all, make
co-initial. Assume for the moment that
are non-coplanar. Complete the parallelopiped with
as adjacent edges:
. This is a vector perpendicular to the plane containing
. We have represented it by
. Let the angle between
. What can
represents the area of the parallelogram
(height of parallelopiped h)
(Area of the base parallelogram)
Volume of the parallelopiped
therefore represents the volume of the parallelopiped with
as adjacent edges.
Note that the volume
of the parallelopiped could equally well have been specified as
Thus, we come to an important property of the
that is, if the vectors are cyclically permuted, the value of the
remains the same. However, note that
August 08, 2014
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#Transformation of Complex Functions : Mapping of Z plane in to W plane ...