However, in three and higher dimensions, rotations about different axes do not commute, and the commutation of 2-D rotation matrices does not extend to general dimensionality. In 3-D space, for example, a rotation carries the x, y, and z axes (conventionally defined as a right-handed system) into a right-handed set of orthogonal unit vectors at another orientation, and the effect of the rotation will be encapsulated in a 3 × 3 orthogonal matrix whose rows give the components of the rotated unit vectors in terms of the original coordinates. Nevertheless, the concepts extend to three-dimensional (3-D) space (and formally, to spaces of arbitrary dimensionality). ![]() Our illustration of orthogonal matrices and rotation was restricted to a 2-D example.
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