MATH.9:5.3 - Return a direction after geometric normalization

The input is an unordered pair P={-v,v} of opposite nonzero vectors in the plane. The requested answer is one of the two unit directions along its axis. Rotating the input should rotate the selected direction.

For P0={(-1,0),(1,0)}, a half-turn leaves the pair unchanged and negates every permitted unit direction. No permitted output is fixed, so a rotation-equivariant deterministic direction cannot be selected from this input.

The same failure appears in normalization. Both the identity and a half-turn map P0 to the same standard pair. Selecting (1,0) there and undoing those transformations returns opposite directions. The standard pair did not determine a direction of return.

Returning both directions, {-v/|v|,v/|v|}, is compatible with rotation when that set is the required output. Averaging them returns zero, whose length is zero rather than one.

Alternatively, mark one endpoint w in P and return w/|w|. Every rotation R preserves length, so R*w/|R*w|=R*(w/|w|). The marked-input rule is therefore equivariant. It answers a question with additional supplied information, which the unmarked pair lacked.