Mathematics-Online course: Vector Calculus-Parallelepidial Product-Properties of the Parallelepidial Product

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	Mathematics-Online course: Vector Calculus - Parallelepidial Product
	Properties of the Parallelepidial Product

The parallelepidial product is linear in each argument. In addition, it has the following properties:

Cyclic Permutation:

$\displaystyle [\vec{a},\vec{b},\vec{c}] = [\vec{b},\vec{c},\vec{a}] = [\vec{c},\vec{a},\vec{b}]$
Linear Dependence:

$\displaystyle [\vec{a},\vec{b},\vec{c}] = 0 \Leftrightarrow \vec{0}=\alpha\vec{a} + \beta\vec{b} + \gamma\vec{c}\,,$

with at least one scalar $\alpha, \beta, \gamma$ beeing unequal to 0 .
Orientation:

$\displaystyle [\vec{a},\vec{b},\vec{c}] > 0$

for every system of three vectors oriented by the Right-Hand-Rule.

automatically generated 10/30/2007