triple cross product

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

The of the triple cross product or Lagrange’s is


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

The the vectors b and c (when these are not parallel).

Note that the use of parentheses in the triple cross products is necessary, since the cross product operationMathworldPlanetmath is not associative (, i.e., generally we have


(for example:  (i×i)×j=0  but  i×(i×j)=-j  when (i,j,k) is a right-handed orthonormal basisMathworldPlanetmath of 3).  So the system (  (3,+,×)  is not a ring.

A direct consequence of the is the Jacobi identityMathworldPlanetmath


which is one of the properties making  (3,+,×)  a Lie algebra.

It follows from the also that


where (uvw) means the triple scalar product of u, v and w.

Title triple cross product
Canonical name TripleCrossProduct
Date of creation 2013-03-22 14:15:53
Last modified on 2013-03-22 14:15:53
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 28
Author pahio (2872)
Entry type Definition
Classification msc 15A72
Synonym vector triple product
Synonym triple vector product
Related topic PhysicalVector
Defines Lagrange’s formulaMathworldPlanetmathPlanetmath