# 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

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

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

The the vectors $\vec{b}$ and $\vec{c}$ (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

 $(\vec{a}\times\vec{b})\times\vec{c}\neq\vec{a}\times(\vec{b}\times\vec{c})$

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

A direct consequence of the is the

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

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

It follows from the also that

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

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

Title triple cross product TripleCrossProduct 2013-03-22 14:15:53 2013-03-22 14:15:53 pahio (2872) pahio (2872) 28 pahio (2872) Definition msc 15A72 vector triple product triple vector product PhysicalVector Lagrange’s formula