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I need a vector function for finding two position vectors which are each on the same ray, and which each orbit on concentric spheres around a given center $\{cx,cy, cz\}$

The first position vector is on a circle with radius $r$, and the second position vector is on a sphere with radius $r+1$.

Given a starting point at the point latitude $0$, longitude $0$, I want the function to allow both vectors to move in synchronous orbits along the line latitude $0$.

The origin is the point $(0, 0, 0)$, which is also a point on the outermost concentric circle. Similarly, the corresponding starting point on the inner concentric circle is $(0, 0, 1)$. Thus, $\{cx,cy,cz\}$ are a distance in front of the origin, and the origin is a point on the outermost sphere.

I do not have any experience with this. Can someone show me how to set this up? It took a great deal of work and research for me to frame this problem to this step. I have done well at 3 semesters of calculus, but I have not taken linear algebra yet. This problem is for my own learning. I am writing a computer program for fun, and it would be nice to see a function that looks easy to program.

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