ONE MORE STEP TOWARD BETTER TOMORROW : CHAPTER 24 -Scalar Triple Product

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

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

Friday, 8 August 2014

CHAPTER 24 -Scalar Triple Product

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