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Vector Triple Product

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

{\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)

See also BAC-CAB Rule, Cross Product, Dot Product, Levi-Civita Symbol, Scalar Triple Product, Vector Quadruple Product


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.

© 1996-9 Eric W. Weisstein