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After obtaining the 2D rotation matrix (as a function of rotation angle) once by geometry and once by complex algebra, I tried to obtain it by invariance of the Euclidean metric. By this approach, the four elements of the 2D rotation matrix can be determined in terms of a single adjustable parameter, which is all right as rotation in 2D is characterised by a single parameter.

However when I try this with the 3D rotation matrices, I get 3 adjustable parameters. But 3D rotation is characterised by only 2 independent parameters! What's happening?

Further, I can show that the 2D rotation matrices must have determinant equal to 1/-1. Is the same true for 3 or higher dimensions?

Thanks.

Molu