triple cross product TripleCrossProduct 2004-03-16 13:55:08 2006-03-15 03:34:31 Definition cross product Lagrange's formula % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The cross product of a vector with a cross product is called the {\em triple cross product}. The \PMlinkescapetext{{\em expansion formula}} of the triple cross product or {\em Lagrange's \PMlinkescapetext{formula}} 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 \PMlinkescapetext{factor}). The \PMlinkescapetext{formula shows that this vector is in the plane spanned by} 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 \PMlinkname{associative}{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 \PMlinkname{system}{AlgebraicSystem}\, $(\mathbb{R}^3,\,+,\,\times)$\, is not a ring. A direct consequence of the \PMlinkescapetext{expansion formula} is the {\em Jacobi identity} $$\vec{a}\times(\vec{b}\times\vec{c})+\vec{b}\times(\vec{c}\times\vec{a})+ \vec{c} \times(\vec{a}\times\vec{b}) = \vec{0},$$ which is one of the properties making\, $(\mathbb{R}^3,\,+,\,\times)$\, a Lie algebra. It follows from the \PMlinkescapetext{expansion formula} 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 $\vec{w}$.