Rotations in 3-space

Rotations in 3-space are more complicated than rotations in a plane. One reason for this is that rotations in 3-space in general do not commute. (Give an example!) As Feynman says, we don't have a good intuition about rotations in space, because we rarely encounter them in our daily experience. So it is hard for us to imagine what will happen if we rotate a body with respect to one axis by a certain angle, then with respect to another axis by another angle and so on. It would be probably different if we were fish or birds. We consider rotations with fixed center which we place at the origin. Then rotations are linear transformations of the space represented by orthogonal matrices with determinant 1. Orthogonality means that all distances and angles are preserved. The determinant of an orthogonal matrix is always 1 or-1. (Think why.) The condition that determinant equals 1 means that orientation is preserved. An orthogonal transformation with determinant-1 will map a right shoe onto a left shoe; such orthogonal transformations are called sometimes improper rotations, we cannot actually perform them in real life. (No matter how you rotate a left shoe it will never become a right shoe). We begin with simple properties of rotation matrices. Theorem 1 Let A be a rotation matrix. Then 1 is an eigenvalue of A. Two other eigenvalues are either −1 or complex conjugate numbers of absolute value 1. The first statement means that there is a vector x = 0 which remains fixed, Ax = x. So each vector of the one-dimensional subspace spanned by this x remains fixed, that is every rotation has an axis. This is not so in dimension 2. A (non-identity) rotation in dimension 2 displaces every non-zero vector. Exercise: give an example of rotation in dimension 4 which moves every non-zero vector. Proof of Theorem 1. First notice that all eigenvalues of an orthogonal matrix have absolute value 1. Indeed, Ax = λx, Ax 2 = |λ| 2 x 2 , and Ax 2 = x 2 , so |λ| = 1.