01
Translate first and then rotate about the origin
Applying T to (2,1) gives (5,1). Applying R to that intermediate point swaps the coordinate roles and negates the original y value, giving (-1,5). Record the complete path as (2,1) to (5,1) to (-1,5). The translation changes the point before the rotation acts, so the rotation uses five as its new vertical coordinate. It does not rotate the original point and then attach the translation afterward. Keeping the intermediate pair visible makes that distinction explicit and gives a reader a straightforward way to check the arithmetic.
02
Restart and reverse the order of the operations
Return to the original point (2,1). Applying R first gives (-1,2). Applying T next adds three to the horizontal coordinate, giving (2,2). The second path is therefore (2,1) to (-1,2) to (2,2), which differs from the first path’s endpoint (-1,5). Both paths use the same operations once, but their order changes the result. This example establishes that these particular transformations do not commute. It does not claim that every pair of spatial operations behaves this way; other choices of operations or parameters need their own comparison.
03
Undo each path in reverse order to verify it
The inverse of T subtracts three from x. The inverse of R maps (x,y) to (y,-x). To undo the first path, rotate (-1,5) back to (5,1), then subtract three from x to recover (2,1). To undo the second path, subtract three from (2,2) to reach (-1,2), then rotate back to (2,1). Save the starting point, definitions, operation order, intermediate point and endpoint in a reusable table. Reverse-order inverse checks provide a second calculation while preserving the original sequence for inspection.