[?] Represent a vector moving on three mutually perpendicular circles?

Questions
1

Dear Friends,
I have three concentric circles in planes mutually perpendicular to each other (say, x2 + y2 = r2, z2 + y2 = r2, x2 + z2 = r2). There is a vector R (theta, phi), which is constrained to move only on the circumference of these circles (having its tail fixed at the common center of the three circles). How can I represent this vector mathematically in Mathematica, in order to find its dot product with another fixed vector F(theta_1, phi_1).
Will appreciate any suggestion regarding this.
Thanks

2

I hope I have understood you question correctly.
The constriction onto those circles means that your vector R (I will call it below vR) lives on a sphere with radius r with the constrains x==0 or y==0 or z==0. Therefore it make sense to change to cartesian coordinates first. My short code should be self-explanatory:

vR[\[Phi]_, \[Theta]_] := CoordinateTransform["Spherical" -> "Cartesian", {r, \[Phi], \[Theta]}]
vF[\[Phi]_, \[Theta]_] := CoordinateTransform["Spherical" -> "Cartesian", {1, \[Phi], \[Theta]}]
constrains = ConstantArray[1, {3, 3}] - IdentityMatrix[3];
vF[\[Phi]1, \[Theta]1].(vR[\[Phi], \[Theta]] #) & /@ constrains

The result are three solutions, one for each constraint.

Hope that helps, regards – Henrik