|MadSci Network: Physics|
Hi, Good question! Basically rotations are things you do to a vector, and can be described by square matrices. If we have a two dimensional matrix, then any operator on it must be a 2x2 matric. All the entries of these matrices are real numbers. For the operation to be a rotation, we require that the vector doesn't change length, and that all vectors orthogonal vectors are mapped on to orthogonal vectors. Basically this means that the vectors which are at right angles to each other must still be at right angles after the roataion. The columns in the matrix correspond to the vectors that the basis vectors (like [1 0 0]' [0 1 0]' and [0 0 1]') will be mapped to. In n dimensions, the restriction that we cannot change the length of a vector means that we can only choose n-1 entries in the first column of the nxn matrix. Since the next column has two restrictions (1. it must maintain the length of the vector, and 2. that it must be at right angles to the first column), we can only choose n-2 entries freely. For column m we have m restrictions, and so can only choose (n-m) entries. So we now add up all the free entries in the matrix, and that gives us the number of rotations we need! So we have (n-1)(n-2)(n-3)...(1) = n(n-1)/2. For 1 dimension this is 0, for 2d this is 1, for 3d it is 3, so for 4d it is 6 and for 5d it is 10. I should point out that in 3d you can actually get any rotation you want from rotations about only two axes. You can do this by rotating about the X axis, then the Y axis and then the X axis again. These are called Euler angles, and I've added a reference for them at the bottom if you want to learn more. Hope this helps! Joe Some references you may want to check out: http://mathworld.wolfram.com/RotationMatrix.html http://mathworld.wolfram.com/EulerAngles.html
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