Orthogonality

The vectors (signals) and ^5.11are said to be orthogonal if , 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 two perpendicular lines is , and , as expected. More generally, orthogonality corresponds to the fact that two vectors in -space intersect at a right angle and are thus perpendicular geometrically.

Example ():

Let and , as shown in Fig.5.8.

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.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.