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Parallelepidial Product - Calculation of Coordinates

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If the vectors $ \vec{u}$ , $ \vec{v}$ and $ \vec{w}$ span a proper parallelepiped, then every vector $ \vec{x}$ can be represented by a linear combination

$\displaystyle \vec{x} = \alpha \vec{u} + \beta \vec{v} + \gamma \vec{w} $

with the coefficients

$\displaystyle \alpha = \frac{[\vec{x},\vec{v},\vec{w}]} {[\vec{u},\vec{v},\vec... ... \gamma = \frac{[\vec{x},\vec{u},\vec{v}]} {[\vec{w},\vec{u},\vec{v}]}\, . $

see also:

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  automatically generated 3/28/2008