Example:

For $N=3$ we have, in general,

$$\left<x,y\right> = x_0 \overline{y_0} + x_1 \overline{y_1} + x_2 \overline{y_2}.$$

Let

$$\begin{eqnarray*} x &=& [0,j,1] \\ y &=& [1,j,j]. \end{eqnarray*}$$

Then

$$\left<x,y\right> = 0\cdot 1 + j \cdot (-j) + 1 \cdot (-j) = 0 + 1 + (-j) = 1-j.$$