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Scalar Product of Real Vectors

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For vectors $ v = (v_1,\dots,v_n), w = (w_1,\dots,w_n) \in \mathbb{R}^n$ the scalar product is defined by

$\displaystyle \langle v,w \rangle := \sum_{i=1}^n v_i w_i = v_1w_1 + \dots + v_nw_n $

and the associated norm is

$\displaystyle \vert v\vert = \sqrt{v_1^2 + \dots + v_n^2}\, . $

\includegraphics{Def_Skalarprod.eps}

Geometrically the scalar product can be defined by

$\displaystyle \langle v,w \rangle := \vert v\vert\vert w\vert\cos \alpha $

where $ \alpha$ is the smaller one of the two angles between $ v$ and $ w$. That is, the scalar product is the oriented length of the projection of one vector onto the other one multiplied by the magnitude of the vector onto which the former was projected.

see also:

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  automatically generated 2/10/2005