Vector Quadruple Product

$$({\bf A}\times {\bf B})\cdot ({\bf C}\times {\bf D}) = ({\bf... ... B}\cdot {\bf D})-({\bf A}\cdot {\bf D})({\bf B}\cdot {\bf C})$$ (1)

$$({\bf A}\times {\bf B})^2 \equiv ({\bf A}\times {\bf B})\cdot({\bf A}\times {\bf B})$$

$$= ({\bf A}\cdot {\bf A})({\bf B}\cdot {\bf B})-({\bf A}\cdot {\bf B})({\bf B}\cdot {\bf A})$$

$$= {\bf A}^2{\bf B}^2-({\bf A}\cdot {\bf B})^2$$ (2)

$${\bf A}\times ({\bf B}\times ({\bf C}\times {\bf D})) = {\bf... ...\times {\bf D}))-({\bf A}\cdot {\bf B})({\bf C}\times {\bf D})$$ (3)

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

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

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

See also Lagrange's Identity, Vector Triple Product