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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 $\vec b \cdot \vec c$is 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
 $[\vec a\;\;\vec b\;\vec c] = \vec a \cdot (\vec b \times \vec c)$ $= - \vec a \cdot (\vec c \times \vec b)$ $= - [\vec a\;\;\vec c\;\;\vec b]$