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Three-Dimensional Rotation Matrices

Physics 216 Spring 2012. Three-Dimensional Rotation Matrices 1. Rotation Matrices A real orthogonal matrix R is a matrix whose elements are real numbers and satisfies 1. R = RT (or equivalently, RRT = I, where I is the n n identity matrix). Taking the determinant of the equation RRT = I and using the fact that det(RT ) = det R, it follows that (det R)2 = 1, which implies that either det R = 1 or det R = 1. A. real orthogonal n n matrix with det R = 1 is called a special orthogonal matrix and provides a matrix representation of a n-dimensional proper rotation1 ( no mirrors required!). The most general Three-Dimensional Rotation matrix represents a counterclockwise Rotation by an angle about a fixed axis that lies along the unit vector n.

where the axis of rotation and the angle of rotation are specified as arguments of R. The most general three-dimensional rotation, denoted by R(ˆn,θ), can be specified by an axis of rotation, nˆ, and a rotation angle θ. Conventionally, a positive rotation angle corresponds to a counterclockwise rotation. The direction of the axis is deter-

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