Orthogonality

The vectors (signals) $x$ and $y$ are 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: 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.