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Mathematics-Online course: Preparatory Course Mathematics - Linear Algebra and Geometry - Vectors

Parallelepipedial 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\,. $

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  automatically generated 1/9/2017