Mathematics-Online lexicon: Scalar Product

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	Mathematics-Online lexicon:
	Scalar Product

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 scalar product of two vectors is defined by

$\displaystyle \vec{a}\cdot\vec{b}$	$\displaystyle =$	$\displaystyle \vert\vec{a}\vert\vert\vec{b}\vert\cos\sphericalangle(\vec{a},\vec{b})$
	$\displaystyle =$	$\displaystyle a_1 b_1 + a_2 b_2 + a_3 b_3 .$

In particular, we have

$\displaystyle \vec{a}\cdot\vec{a} = \vert\vec{a}\vert^2$

and

$\displaystyle \vec{a}\cdot\vec{b} = 0 \quad \Leftrightarrow \quad \vec{a} \perp \vec{b}\, .$

From the coordinate representation of the scalar product it follows

$\displaystyle \vec{a}\cdot\vec{b} = \vec{b}\cdot\vec{a}$

and

$\displaystyle \left(s\vec{a}+r\vec{b}\right)\cdot\vec{c} = s\vec{a}\cdot\vec{c}+r\vec{b}\cdot\vec{c}\, ,$

i.e., the scalar product has properties analogous to some calculation rules of the product of real numbers.

Annotation:

automatically generated 3/28/2008