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'triple cross product'
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| Title of object: |
triple cross product |
| Canonical Name: |
TripleCrossProduct |
| Type: |
Definition |
| Created on: |
2004-03-16 13:55:08 |
| Modified on: |
2004-12-18 08:36:28 |
| Classification: |
msc:15A72 |
| Synonyms: |
triple cross product=vector triple product |
Revision comment (for changes between this and next version):
| Changes for correction #6762 ('minor'). |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
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%\usepackage{amsthm}
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%\usepackage{xypic}
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Content:
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 the {\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 $\mathbb{R}^3$).
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 $\vec{w}$. |
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