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Orthogonality

The vectors (signals) $ x$ and $ y$5.11are said to be orthogonal if $ \left<x,y\right>=0$, denoted $ x\perp y$. That is to say

$\displaystyle \zbox {x\perp y \Leftrightarrow \left<x,y\right>=0.} $

Note that if $ x$ and $ y$ are real and orthogonal, the cosine of the angle between them is zero. In plane geometry ($ N=2$), the angle between two perpendicular lines is $ \pi/2$, and $ \cos(\pi/2)=0$, as expected. More generally, orthogonality corresponds to the fact that two vectors in $ N$-space intersect at a right angle and are thus perpendicular geometrically.

Example ($ N=2$):

Let $ x=[1,1]$ and $ y=[1,-1]$, as shown in Fig.5.8.

Figure 5.8: Example of two orthogonal vectors for $ N=2$.
\includegraphics[scale=0.7]{eps/ip}

The inner product is $ \left<x,y\right>=1\cdot \overline{1} + 1\cdot\overline{(-1)} = 0$. 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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