Cylindrical Coordinates example

Question

I am trying to describe the surface given by the equation $\theta = \pi/4$. Now, since $x$ and $r$ can be anything, doesn't this describe a vertical plane through the origin, given by $y=x$? I am told that this is the half plane where $x\geq 0$, but I don't see why this is true. If $r$ can be anything, can't it take on negative values?

Answer 1

You forgot that $r$ is the distance to the $zz$ axis. Therefore, it is not true that it can be anyting. It must be non-negative.