Mathematics-Online lexicon: Properties and Rules for Vector Products

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	Mathematics-Online lexicon:
	Properties and Rules for Vector Products

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

The vector product has the following properties:

Antisymmetry

$\displaystyle \vec{a} \times \vec{b} = -\bigl(\vec{b} \times \vec{a}\bigr)$
Linearity

$\displaystyle \bigl(\alpha_1 \vec{a}_1 + \alpha_2 \vec{a}_2\bigr) \times \bigl(\beta_1 \vec{b}_1 + \beta_2 \vec{b}_2\bigr) =\;$ $\displaystyle \alpha_1\beta_1\bigl(\vec{a}_1\times\vec{b}_1\bigr) + \alpha_1\beta_2\bigl(\vec{a}_1\times\vec{b}_2\bigr) +$

$\displaystyle \alpha_2\beta_1\bigl(\vec{a}_2\times\vec{b}_1\bigr) + \alpha_2\beta_2\bigl(\vec{a}_2\times\vec{b}_2\bigr)$
Grassmann Identity

$\displaystyle (\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a}\cdot\vec{c})\,\vec{b}- (\vec{b}\cdot\vec{c})\,\vec{a}$
Lagrange Identity

$\displaystyle (\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d}) = (\vec{a}\cdot\vec{c})(\vec{b}\cdot\vec{d}) - (\vec{a}\cdot\vec{d})(\vec{b}\cdot\vec{c})$

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Annotation:

automatically generated 3/17/2011

$\displaystyle \bigl(\alpha_1 \vec{a}_1 + \alpha_2 \vec{a}_2\bigr) \times \bigl(\beta_1 \vec{b}_1 + \beta_2 \vec{b}_2\bigr) =\;$	$\displaystyle \alpha_1\beta_1\bigl(\vec{a}_1\times\vec{b}_1\bigr) + \alpha_1\beta_2\bigl(\vec{a}_1\times\vec{b}_2\bigr) +$
	$\displaystyle \alpha_2\beta_1\bigl(\vec{a}_2\times\vec{b}_1\bigr) + \alpha_2\beta_2\bigl(\vec{a}_2\times\vec{b}_2\bigr)$