triple cross product

The cross product of a vector with a cross product is called the triple cross product.

The of the triple cross product or Lagrange’s is

$$\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=(\overrightarrow{a}\cdot \overrightarrow{c})\overrightarrow{b}-(\overrightarrow{a}\cdot \overrightarrow{b})\overrightarrow{c}$$

(“exterior dot far times near minus exterior dot near times far” — this works also when “exterior” is the last ).

The the vectors $\overrightarrow{b}$ and $\overrightarrow{c}$ (when these are not parallel).

Note that the use of parentheses in the triple cross products is necessary, since the cross product operation is not associative (http://planetmath.org/GeneralAssociativity), i.e., generally we have

$$(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}\ne \overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})$$

(for example:  $(\overrightarrow{i}\times \overrightarrow{i})\times \overrightarrow{j}=\overrightarrow{0}$  but  $\overrightarrow{i}\times (\overrightarrow{i}\times \overrightarrow{j})=-\overrightarrow{j}$  when$(\overrightarrow{i},\overrightarrow{j},\overrightarrow{k})$ is a right-handed orthonormal basis of ${ℝ}^3$).  So the system (http://planetmath.org/AlgebraicSystem)  $({ℝ}^3,+,\times )$  is not a ring.

A direct consequence of the is the Jacobi identity

$$\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})+\overrightarrow{b}\times (\overrightarrow{c}\times \overrightarrow{a})+\overrightarrow{c}\times (\overrightarrow{a}\times \overrightarrow{b})=\overrightarrow{0},$$

which is one of the properties making  $({ℝ}^3,+,\times )$  a Lie algebra.

It follows from the also that

$$(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{c}\times \overrightarrow{d})=(\overrightarrow{a}\overrightarrow{b}\overrightarrow{d})\overrightarrow{c}-(\overrightarrow{a}\overrightarrow{b}\overrightarrow{c})\overrightarrow{d}$$

where $(\overrightarrow{u}\overrightarrow{v}\overrightarrow{w})$ means the triple scalar product of $\overrightarrow{u}$, $\overrightarrow{v}$ and $\overrightarrow{w}$.