Are the locations of the normal vectors on the planes fixed, or are they only defined as normal to the planes?
If the locations of the normal vectors are not fixed then they can be located anywhere and still be normal to the planes. Given the directions of the plane normals then the lines corresponding to the plane normals can be determined which pass through any arbitrary point in space.
The planes are known(I guess that is fixed). That doesn’t mean you have to use them in the solution(I don’t know if you need them or not). Rotating any of those planes should give a different answer if that helps(as if you imagine the black line, it would move, no longer being at the same point)
edit: reading your question again, yes, they are fixed(at the origin of the plane). Moving any plane would give a different solution.
edit2: well, moving a plane would not guarantee a different solution. For example. If you rotate any of those planes in 3d about the unknown point, this would not change the result.
I don’t know whether I got your problem, but perhaps this is helpful: Closest Points on Two Skew Lines closest_points_on_two_skew_lines.gh (7.1 KB)
This is sounding like a multi-dimensional optimisation/minimisation problem. This may be completely off the mark but could be worth a look for a path to a possible solution…