# The Axis of a Rotation: Analysis, Algebra, Geometry

by Dan Kalman

Mathematics Magazine
October, 1989

Subject classification(s): Linear Algebra
Applicable Course(s): 3.8 Linear/Matrix Algebra

Given any 3 by 3 rotation matrix $A$ (i.e. orthogonal with determinant $1$ and an arbitrary vector $x$, the vector $Ax +A^{T}x+[1−$tr$(A)]x$ lies on the axis of rotation. The article provides three different approaches, requiring various levels of background knowledge, to prove and/or explain the given result.

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Capsule Course Topic(s):
Linear Algebra | Eigenvalues and Eigenvectors
Linear Algebra | Geometry
Linear Algebra | Inner Product Spaces
Linear Algebra | Linear Transformation