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Mathematics-Online course: Linear Algebra - Basic Structures - Scalar Product and Norm

Angles

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For a real scalar product an angle in $ [0,\pi]$ between $ u$ and $ v$ is defined by

$\displaystyle \cos\sphericalangle(u,v) := \frac{\langle u,v\rangle}{\vert u\vert\vert v\vert}\, . $

The definition of angle is motivated by the law of cosines for vectors in $ \mathbb{R}^2$:

\includegraphics[width=7cm]{cosinussatz}

In a triangle the following relation holds true for the lenghts of its sides:

$\displaystyle c^2=a^2+b^2-2ab\cos\left(\angle (AOB)\right)\,. $

On the other hand, since according to the definition of the scalarproduct

$\displaystyle \left\vert\overrightarrow{OA}-\overrightarrow{OB}\right\vert^2 = ... ...t^2 - 2\left\langle\overrightarrow{OA},\overrightarrow{OB}\right\rangle\, , $

the definition of angle results from comparison.

By the Cauchy-Schwarz inequality we obtain for any scalar product

$\displaystyle \frac{\langle u,v \rangle}{\vert u\vert \vert v\vert} \in [-1,1]\, . $

Thus we can equate the value of the quotient with the cosine of an angle in $ [0,\pi]$.

(Authors: Burkhardt/Höllig/Hörner)
  automatically generated 4/21/2005