Vector Triple Product

The triple product can be written in terms of the Levi-Civita Symbol $\epsilon_{ijk}$ as

$\begin{displaymath} {\bf A}\cdot({\bf B}\times{\bf C}) = \epsilon_{ijk} A^iB^jC^k. \end{displaymath}$

(1)

The BAC-CAB Rule can be written in the form

$\displaystyle {\bf A}\times({\bf B}\times{\bf C})$	$\textstyle =$	$\displaystyle {\bf B}({\bf A}\cdot{\bf C})-{\bf C}({\bf A}\cdot{\bf B})$	(2)
$\displaystyle ({\bf A}\times{\bf B})\times{\bf C}$	$\textstyle =$	$\displaystyle -{\bf C}\times({\bf A}\times{\bf B})$
	$\textstyle =$	$\displaystyle -{\bf A}({\bf B}\cdot{\bf C})+{\bf B}({\bf A}\cdot{\bf C}).$	(3)

Additional identities are

${\bf A}\cdot ({\bf B}\times {\bf C}) = {\bf B}\cdot ({\bf C}\times {\bf A}) = {\bf C}\cdot ({\bf A}\times {\bf B})$ (4)

$[{\bf A},{\bf B},{\bf C}]{\bf D}= [{\bf D},{\bf B},{\bf C}]{\bf A}+[{\bf A},{\bf D},{\bf C}]{\bf B}+[{\bf A},{\bf B},{\bf D}]{\bf C}$ (5)

$[{\bf q},{\bf q}',{\bf q}''][{\bf r},{\bf r}',{\bf r}''] = \left\vert\matrix{{\... ...cdot {\bf r}& {\bf q}''\cdot {\bf r}' & {\bf q}''\cdot{\bf r}''\cr}\right\vert.$ (6)

References

Arfken, G. ``Triple Scalar Product, Triple Vector Product.'' §1.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 26-33, 1985.

${\bf A}\cdot ({\bf B}\times {\bf C}) = {\bf B}\cdot ({\bf C}\times {\bf A}) = {\bf C}\cdot ({\bf A}\times {\bf B})$	(4)
$[{\bf A},{\bf B},{\bf C}]{\bf D}= [{\bf D},{\bf B},{\bf C}]{\bf A}+[{\bf A},{\bf D},{\bf C}]{\bf B}+[{\bf A},{\bf B},{\bf D}]{\bf C}$	(5)
$[{\bf q},{\bf q}',{\bf q}''][{\bf r},{\bf r}',{\bf r}''] = \left\vert\matrix{{\... ...cdot {\bf r}& {\bf q}''\cdot {\bf r}' & {\bf q}''\cdot{\bf r}''\cr}\right\vert.$	(6)