Vector Quadruple Product

$\begin{displaymath} ({\bf A}\times {\bf B})\cdot ({\bf C}\times {\bf D}) = ({\bf... ... B}\cdot {\bf D})-({\bf A}\cdot {\bf D})({\bf B}\cdot {\bf C}) \end{displaymath}$

(1)

$\displaystyle ({\bf A}\times {\bf B})^2$	$\textstyle \equiv$	$\displaystyle ({\bf A}\times {\bf B})\cdot({\bf A}\times {\bf B})$
	$\textstyle =$	$\displaystyle ({\bf A}\cdot {\bf A})({\bf B}\cdot {\bf B})-({\bf A}\cdot {\bf B})({\bf B}\cdot {\bf A})$
	$\textstyle =$	$\displaystyle {\bf A}^2{\bf B}^2-({\bf A}\cdot {\bf B})^2$	(2)

$\begin{displaymath} {\bf A}\times ({\bf B}\times ({\bf C}\times {\bf D})) = {\bf... ...\times {\bf D}))-({\bf A}\cdot {\bf B})({\bf C}\times {\bf D}) \end{displaymath}$

(3)

$\displaystyle ({\bf A}\times {\bf B})\times ({\bf C}\times {\bf D})$	$\textstyle =$	$\displaystyle [{\bf A},{\bf B},{\bf D}]{\bf C}-[{\bf A},{\bf B},{\bf C}]{\bf D}$
	$\textstyle =$	$\displaystyle ({\bf C}\times {\bf D})\times ({\bf B}\times {\bf A}) = [{\bf C},{\bf D},{\bf A}]{\bf D}-[{\bf C},{\bf D},{\bf B}]{\bf A},$	(4)

where $[{\bf A},{\bf B},{\bf D}]$ denotes the Vector Triple Product. Equation (24) is known as Lagrange's Identity.