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Scalar Triple Product

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 multiplied by 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 Figure A.106. This volume is independent of how the triple product is formed from ${\bf a}$ , ${\bf b}$ , and ${\bf c}$ , except that

$\begin{displaymath} {\bf a} \cdot {\bf b}\times{\bf c} = - {\bf a} \cdot {\bf c}\times{\bf b}. \end{displaymath}$

(1310)

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 a right-hand circulation 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,

$\begin{displaymath} {\bf a} \cdot {\bf b}\times{\bf c} = {\bf a} \times{\bf b} \cdot{\bf c}. \end{displaymath}$

(1311)

The triple product is also invariant under any cyclic permutation of ${\bf a}$ , ${\bf b}$ , and ${\bf c}$ ,

$\begin{displaymath} {\bf a} \cdot {\bf b} \times{\bf c} = {\bf b} \cdot {\bf c} \times{\bf a} = {\bf c} \cdot {\bf a} \times{\bf b}, \end{displaymath}$

(1312)

but any anti-cyclic permutation causes it to change sign,

$\begin{displaymath} {\bf a} \cdot {\bf b} \times{\bf c} = - {\bf b} \cdot {\bf a} \times{\bf c}. \end{displaymath}$

(1313)

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 coplanar.

**Figure A.106:** *A vector parallelepiped.*
$\begin{figure} \epsfysize =1.5in \centerline{\epsffile{AppendixA/figA.09.eps}} \end{figure}$

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

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

(1314)

Forming the dot product of this equation with ${\bf b}\times {\bf c}$ , we then obtain

$\begin{displaymath} {\bf r} \cdot {\bf b} \times{\bf c} = \alpha\, {\bf a}\cdot{\bf b} \times{\bf c}, \end{displaymath}$

(1315)

$\begin{displaymath} \alpha = \frac{{\bf r}\cdot{\bf b}\times{\bf c}}{{\bf a}\cdot{\bf b}\times{\bf c}}. \end{displaymath}$

(1316)

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 vectors are non-coplanar.

Next: Vector Triple Product Up: Vector Algebra and Vector Previous: Rotation

Richard Fitzpatrick 2011-03-31