Abstract
One application of elementary linear algebra that never fails to surprise my students is that a composition of rotations in space is itself a rotation. The more general result that any 3 by 3 orthogonal matrix of determinant one (a proper orthogonal transformation) is a rotation can be demonstrated most convincingly if one has a computer package which will do the messy arithmetic involved. In this lesson we find the axis and angle of rotation of such a transformation. It brings together such different concepts from linear algebra as orthogonal bases and matrices, the geometric interpretation of eigenvalues and eigenvectors (both real and complex), and the geometric meaning of changing bases. We presume that the students have seen the 2 by 2 matrix of a rotation in the plane and that they know a little about orthogonal matrices and eigenvalues and eigenvectors.
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