How to find the coordinates of one coordinate system in another along a given line?

Question

Mathematics

Suppose there are two Cartesian coordinate systems and there is a straight line whose equation we know in both of these systems. How to find the relative position of coordinate systems (i.e., for example, the coordinates of the origin of the second system in the first, as well as the angle of rotation)?
PS Both the case of the plane and space are interesting.

Answer 1

For simplicity, consider a special case: the line passes through the origin in both the 1st and 2nd systems, the systems are rotated in the same way. It turns out that the equations of the straight line in the two systems completely coincide, and the shift between the systems can be anything. Those. impossible to determine unambiguously, there will be infinitely many solutions.