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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: |
2004-03-16 15:36:53 |
| Classification: |
msc:15A72 |
| Synonyms: |
triple cross product=vector triple product |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
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%\usepackage{psfrag}
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Content:
The cross product of a vector with a cross product is called the {\em triple cross product}.
The {\em expansion formula}
$$\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b}-(\vec{a} \cdot \vec{b})\vec{c}$$
of the triple cross product shows that this triple product is in the plane spanned by the vectors $\vec{b}$ and $\vec{c}$.
Note that the use of parentheses in the triple cross products is necessary, since the cross product operation is not associative, 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 an orthonormal basis of $\mathbb{R}^3$). |
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