Vector Cosine

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Vector Cosine

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K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 © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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Vector Cosine

The Cauchy-Schwarz Inequality can be written

\

In the case of real vectors $\, we can always find a real number$\ which satisfies

\

We thus interpret $\ as the angle between two vectors in ${\. In ${\ we can similarly define $\.

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