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

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

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 $N=2$ .

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.