In limited experiments I observe an equivalence and would like to know if it is proven to be true.
Suppose we have a 3D coordinate system, where we label the coordinates, held to be the original ones, x, y, and z, and we label new coordinates that may result from any rotation on any axis x', y', and z'. I find this equivalence:
r.y + r.x + r.z = r.z + r.x' + r.y'
In words: stacking rotations on the original coordinates produces the same result as reversing the order of the axes that are rotated, starting with an original axis, then rotating by a new axis, and again on a new axis.
Has this been proven true? If yes, does the proof have a name, a subject that I could look up?
The reason I'm interested in this is because as far as I understand how rotation matrices are constructed, they work like the right side of the equation above. The rotation matrix encodes a rotation on an original axis, then a new one, then a new one. It seems to me that it may be helpful in some situations to know that the same result may be achieved by reversing the rotation order and using original axes only.