BAC-CAB Identity

The Vector Triple Product identity

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

This identity can be generalized to $n$-D

$${\bf a}_2\times \cdots\times {\bf a}_{n-1}\times({\bf b}_1\t... ...1 & \cdots & {\bf a}_{n-1}\cdot {\bf b}_{n-1}\cr}}\right\vert.$$

See also Lagrange's Identity