One of the practice problems I was doing is:
Find the vector equation of the surface of cylinder with radius $a>0$ and the axis that goes through the origin and has the direction of vector $\overrightarrow{e}$.
The solution is: $|\overrightarrow{r} \times \overrightarrow{e}|=a$, but I have no idea how to derive it. I would appreciate if someone would explain to me how to get that (I was thinking about it a lot, but it is completely different from a vector equation of a line and I have no idea what to do).
I would also like to know how to get the more general vector equation of a cylinder, whose axis does not go through the origin. And I was thinking if I could perhaps derive the vector equation of a circle in 2D space, but I had no success with that either, so I would also appreciate any help with that.
Thank you in advance!