Hi, gralla55.

I don't know if I understand things perfectly, but I will respond to what I think you mean; you can judge for yourself if what I've said is of any use or not.

First, there might be some confusion because the point (2,1,1) is a point in 3-dimensional space and polar coordinates are used to describe points in 2-dimensional space (i.e. the plane). If we wanted to use something besides Cartesian coordinates to describe the point (2,1,1), we could use something like cylindrical or spherical coordinates.

Next, when giving the coordinates of a point in two different coordinate systems we will (generally) not have the same numbers for the individual coordinates, but both sets of coordinates will give the location of the same point.

I think an example would do us some good.

Say we are considering the point (0,1) in Cartesian coordinates in the plane (this is the point that sits one unit above the origin). The polar coordinates of this point would be (let me know if it's unclear why this is so). We notice that neither coordinates "match up" (so to speak), but both describe the same point.

Perhaps what you're after is the conversion that allows us to convert a point from polar coordinates to Cartesian coordinates. These are:

Notice that if we plug and in the above equations we end up with our original Cartesian coordinates (0,1).

Not sure if this is what you're after, but I thought I'd take a stab in the dark. If there's still some confusion, let me know and we can revist the topic

Good luck!