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If a solid ball (like the Earth) is rotating in 3D space, you can point a single 3D vector out of the North Pole (according to the right hand rule), with the length of that vector proportional to the speed of rotation, and that single vector unambiguously and completely describes how the object is rotating.

Earth with rotation vector

My question is: does this extend to 4 dimensions and higher? Or is more than 1 vector needed to completely pin down how a solid object rotates in 4D? If so, how does the number of vectors needed increase with dimension?

chausies
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  • Think about the eigen values and eigen vectors of the associated rotation matrix and you will have your answer. – Tpofofn Jan 04 '24 at 01:32
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    For the higher dimensional analogues of rotations you want the special orthogonal group. The wikipedia page can answer your questions https://en.wikipedia.org/wiki/Orthogonal_group – Ethan Bolker Jan 04 '24 at 01:33
  • The use of a vector to represent a rotation in 3D has a lot to do with the fact that any 3D rotation leaves exactly a one-dimensional subspace fixed, which coincidentally can be uniquely (up to a single sign flip) identified by a unit vector, which one can then scale to represent the angle of rotation. In $n>3$ dimensions you have rotations that leave an $(n-2)$-dimensional subspace fixed, which doesn't have such a nice mapping to vectors (yes, you can choose a spanning set, but which of the infinite number of choices, and what about all the vector magnitudes?). – David K Jan 04 '24 at 02:28

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