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

Posted 1 year ago
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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(theta1, phi1). Will appreciate any suggestion regarding this. Thanks


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

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