Mathematics-Online lexicon: Parallelepidial Product

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	Mathematics-Online lexicon:
	Parallelepidial Product

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The parallelepidial product, except for the sign,

$\displaystyle \bigl[\vec{a},\vec{b},\vec{c}\bigr] = \vec{a}\cdot(\vec{b}\times\vec{c}) = a_1(b_2c_3-b_3c_2)+a_2(b_3c_1-b_1c_3)+a_3(b_1c_2-b_2c_1)$

yields the volume of the parallelepiped spanned by the three vectors $\vec{a}$ , $\vec{b}$ , $\vec{c}$ .

$\includegraphics[width=7.4cm]{spatprodukt.eps}$

Using the $\varepsilon$ -tensor the parallelepidial product can be expressed as

$\displaystyle \bigl[\vec{a},\vec{b},\vec{c}\bigr]=\sum_{i,j,k=1}^3 \varepsilon_{i,j,k} a_i b_j c_k\,.$

Annotation:

automatically generated 3/28/2008