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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 \vec a,\;\vec b and \vec c. Consider the quantity \vec a \cdot (\vec b \cdot \vec c). Since \vec b \cdot \vec cis a scalar, you cannot define its dot product with another vector. Thus, \vec a \cdot (\vec b \cdot \vec c) is a meaningless quantity.
However, consider the expression \vec a \cdot (\vec b \times \vec c). Since \vec b \times \vec c is a vector, its dot product with \vec a is defined. Thus, \vec a \cdot (\vec b \times \vec c) is defined and is termed the scalar triple product of \vec a,\,\,\vec b and \vec c. This product is represented concisely as [\vec a\;\;\vec b\;\;\vec c].
An alert reader might have noticed that another valid triple product is possible: \vec a \times (\vec b \times \vec c). 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 (STP) [\vec a\;\;\vec b\;\;\vec c].
First of all, make \vec a,\;\;\vec b,\;\;\vec c co-initial. Assume for the moment that \vec a,\;\;\vec b,\;\;\vec c are non-coplanar. Complete the parallelopiped with \vec a,\;\;\vec b,\;\;\vec c as adjacent edges:
Consider \vec b \times \vec c. This is a vector perpendicular to the plane containing \vec b and \vec c. We have represented it by \overrightarrow {OE} . Let the angle between \vec a and \overrightarrow {OE}  be \theta . What can \vec a \cdot (\vec b \times \vec c) i.e. \vec a \cdot \overrightarrow {OE}  represent ? \left| {\overrightarrow {OE} } \right| represents the area of the parallelogram OBDC.
Thus,
\vec a \cdot \overrightarrow {OE}  = \left| {\vec a} \right|\;\left| {\overrightarrow {OE} } \right|\cos {\rm{\theta }}
 = (\left| {\vec a} \right|\cos {\rm{\theta }})\;\overrightarrow {OE}
= (height of parallelopiped h) \times (Area of the base parallelogram)
= Volume of the parallelopiped
The STP\;\,[\vec a\;\vec b\;\vec c] therefore represents the volume of the parallelopiped with \vec a,\;\vec b,\;\vec c as adjacent edges.
Note that the volume V of the parallelopiped could equally well have been specified as
V = \vec b \cdot (\vec c \times \vec a) = [\vec b\;\vec c\;\vec a]
 = \vec c \cdot (\vec a \times \vec b) = [\vec c\;\vec a\;\vec b]
Thus, we come to an important property of the STP:
[\vec a\;\vec b\;\vec c] = [\vec b\;\vec c\;\vec a] = [\vec c\;\vec a\;\vec b]
that is, if the vectors are cyclically permuted, the value of the STP remains the same. However, note that
[\vec a\;\;\vec b\;\vec c] = \vec a \cdot (\vec b \times \vec c)
 =  - \vec a \cdot (\vec c \times \vec b)
 =  - [\vec a\;\;\vec c\;\;\vec b]
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