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Scalar Triple Product

 

The Scalar Triple Product

Figure 1: The scalar triple product.

\begin{figure} \epsfysize =2in \centerline{\epsffile{fig9.eps}} \end{figure}
Consider three vectors ${\bf a}$, ${\bf b}$, and ${\bf c}$. The scalar triple product is defined ${\bf a}\cdot {\bf b}\times {\bf c}$. Now, ${\bf b}\times {\bf c}$ is the vector area of the parallelogram defined by ${\bf b}$ and ${\bf c}$. So, ${\bf a}\cdot {\bf b}\times {\bf c}$ is the scalar area of this parallelogram times the component of ${\bf a}$ in the direction of its normal. It follows that ${\bf a}\cdot {\bf b}\times {\bf c}$ is the volume of the parallelepiped defined by vectors ${\bf a}$, ${\bf b}$, and ${\bf c}$ (see Fig. 1). This volume is independent of how the triple product is formed from ${\bf a}$, ${\bf b}$, and ${\bf c}$, except that
\begin{displaymath} {\bf a} \cdot {\bf b}\times{\bf c} = - {\bf a} \cdot {\bf c}\times{\bf b}. \end{displaymath} (1)

So, the ``volume'' is positive if ${\bf a}$, ${\bf b}$, and ${\bf c}$ form a right-handed set (i.e., if ${\bf a}$ lies above the plane of ${\bf b}$ and ${\bf c}$, in the sense determined from the right-hand grip rule by rotating ${\bf b}$ onto ${\bf c}$) and negative if they form a left-handed set. The triple product is unchanged if the dot and cross product operators are interchanged:
\begin{displaymath} {\bf a} \cdot {\bf b}\times{\bf c} = {\bf a} \times{\bf b} \cdot{\bf c}. \end{displaymath} (2)

The triple product is also invariant under any cyclic permutation of ${\bf a}$, ${\bf b}$, and ${\bf c}$,
\begin{displaymath} {\bf a} \cdot {\bf b} \times{\bf c} = {\bf b} \cdot {\bf c} \times{\bf a} = {\bf c} \cdot {\bf a} \times{\bf b}, \end{displaymath} (3)

but any anti-cyclic permutation causes it to change sign,
\begin{displaymath} {\bf a} \cdot {\bf b} \times{\bf c} = - {\bf b} \cdot {\bf a} \times{\bf c}. \end{displaymath} (4)

The scalar triple product is zero if any two of ${\bf a}$, ${\bf b}$, and ${\bf c}$ are parallel, or if ${\bf a}$, ${\bf b}$, and ${\bf c}$ are co-planar.

If ${\bf a}$, ${\bf b}$, and ${\bf c}$ are non-coplanar, then any vector ${\bf r}$ can be written in terms of them:

\begin{displaymath} {\bf r} = \alpha \,{\bf a} + \beta\,{\bf b} + \gamma\, {\bf c}. \end{displaymath} (5)

Forming the dot product of this equation with ${\bf b}\times {\bf c}$, we then obtain
\begin{displaymath} {\bf r} \cdot {\bf b} \times{\bf c} = \alpha\, {\bf a}\cdot{\bf b} \times{\bf c}, \end{displaymath} (6)

so
\begin{displaymath} \alpha = \frac{{\bf r}\cdot{\bf b}\times{\bf c}}{{\bf a}\cdot{\bf b}\times{\bf c}}. \end{displaymath} (7)

Analogous expressions can be written for $\beta$ and $\gamma$. The parameters $\alpha$, $\beta$, and $\gamma$ are uniquely determined provided $ {\bf a}\cdot{\bf b} \times{\bf c} \neq 0$: i.e., provided that the three basis vectors are not co-planar.

 

 

 

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