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Vector Cosine

The Cauchy-Schwarz Inequality can be written

$\displaystyle \frac{\left\vert\left<\underline{u},\underline{v}\right>\right\vert}{\Vert\underline{u}\Vert\cdot\Vert\underline{v}\Vert} \leq 1.
$

In the case of real vectors $ \underline{u},\underline{v}$ , we can always find a real number $ \theta\in[0,\pi]$ which satisfies

$\displaystyle \zbox {\cos(\theta) \isdef \frac{\left<\underline{u},\underline{v}\right>}{\Vert\underline{u}\Vert\cdot\Vert\underline{v}\Vert}.}
$

We thus interpret $ \theta$ as the angle between two vectors in $ {\bf R}^N$ .


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-21 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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