I know that we can defined different dot product on a vector space. Yet when we are working in an orthogonal basis we have :
$$\langle x,y\rangle = x_1y_1 + \cdots +x_ny_n$$
So does it mean that every dot product is actually the same (because it’s just the product of the coordinates) ?
I know this must be false, but I don’t understand why because the fact that $\langle x,y\rangle = 0$ when $x,y$ are orthogonal means that every dot product is actually the product of the coordinates of the two vectors.