Scalar Triple Product Of Vectors

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 \(\vec a,\,\,\vec b\,\,{\text{and}}\,\vec c\) . 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 \(\vec b\,\,{\text{and}}\,\,\vec c\)  . 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,

\[\begin{align}&\vec a \cdot \overrightarrow {OE}  = \left| {\vec a} \right|\;\left| {\overrightarrow {OE} } \right|\cos {\theta } \hfill \\\\&\qquad\;\;= (\left| {\vec a} \right|\cos {\theta })\;\overrightarrow {OE}  \hfill \\\\&\qquad\;\; = {\text{ }}\left( {Height{\text{ }}of{\text{ }}the{\text{ }}parallelopiped\,h} \right){\text{ }} \times {\text{ }}\left( {Area{\text{ }}of{\text{ }}the{\text{ }}base{\text{ }}parallelogram} \right) \hfill \\\\&\qquad\;\;   = {\text{ }}Volume{\text{ }}of{\text{ }}the{\text{ }}parallelopiped. \hfill \\ \end{align} \]

\[\boxed{\begin{align}&{\text{The}}\;\,STP\;\,[\vec a\;\vec b\;\vec c]\;\,{\text{therefore}}\;\,{\text{represents}}\;\,{\text{the}}\;\,{\text{volume}}\;\, \hfill \\&{\text{of}}\;{\text{the}}\;\,{\text{parallelopiped}}\;{\text{with}}\;\vec a,\;\vec b,\;\vec c\;{\text{as}}\;{\text{adjacent}}\;{\text{edges}} \hfill \\ \end{align}}\]

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

\[\begin{align}&[\vec a\;\;\vec b\;\vec c]= \vec a \cdot (\vec b \times \vec c) \hfill \\\\& \qquad\quad=  - \vec a \cdot (\vec c \times \vec b) \hfill \\\\&\qquad\quad =  - [\vec a\;\;\vec c\;\;\vec b] \hfill \\ \end{align} \]