polar coordinates of a 2 dimensional vector are:
x = r cos θ
y = r sin θ
What will be the polar coordinates of a vector in 3D (x, y, z)?
I don't think the responders to this question and the authors of the literature on the internet really understand what 3D polar coordinates are. The only concise definition I have found thus far has been in the Schaum's Outline Series "Analytic Geometry" on page 145.
For 2D polar coordinates, a radius and one angle are all that is needed to define a point on a plane in polar coordinates. But, for 3D polar, a radius and 3 angles are needed. The first angle is from the x-axis to the radius, the second angle from the y-axis to the radius, and the third angle from the z-axis to the radius. Notice that I said from the coordinate axis to the radius, not the coordinate axis to the projection of the radius on the xy-plane.
All of the discussions I have seen thus far devolve into cylindrical or spherical coordinate systems which measure from the x-axis to the projection of the radius to the xy-plane and don't use the angle from the y-axis to the radius at all.
An example exercise from the Schaum's book cited above is as follows.
Given point is {1,-2,2}
Answer: 3D Polar coordinate are {3,70°32',131°49',48°11'}
Unfortunately, Mathematica is completely AWOL on 3D polar as far as know.