**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW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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## Orthogonality

The vectors (signals) and are said to be

orthogonalif , denoted . That is to say

Note that if and are real and orthogonal, the cosine of the angle between them is zero. In plane geometry (), the angle between twoperpendicular lines is , and , as expected. More generally, orthogonality corresponds to the fact that two vectors in-space intersect at a

right angleand are thusperpendiculargeometrically.

Example ():Let and , as shown in Fig. 6.8.

The inner product is . This shows that the vectors are

orthogonal. As marked in the figure, the lines intersect at a right angle and are therefore perpendicular.