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Option 4, none of the above. Your option 1 is wrong because points don't rotate. Your option 2 is closer to correct, but ultimately still wrong. You're overly hung up on points (the origin). It might help to get a handle on what "rotation" is. Points don't rotate. Better said, a rotated point is indistinguishable from the original. What about one ...

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A very late answer, one that I hope adds to the excellent answers by Mark and Luboš. From the perspective of Newtonian mechanics, there's nothing wrong per se with using a geocentric point of view. Such a point of view does require adding fictitious forces and torques that would otherwise be absent in an inertial perspective, but if makes sense to do that, ...

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It is usual to make a distinction between proper rotation as opposed to all rotations (which include those that change handedness). The matrices of proper rotations have determinate +1 while those of improper rotations have determinate -1. In terms of groups the proper rotations are $\mathrm{SO}(3)$ while all rotations taken together are the orthagonal ...

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[...] my question concerns two relatively moving observers. Can one of them distinguish that the other has adopted the oppositely handed coordinate frame? What information does observer A have about observer B? If all A knows about B is B's state of motion, and if he assumes that B is going to choose coordinates in which he is at rest at the origin, ...

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