The Equivalent of a Series of Rotations

  One can subject the coordinate axes to a series of plane rotations and then try to find an equivalent rotation to produce the same orientation of the axes. Suppose one first moves the x-axis in the xy-plane leaving the z-axis fixed then moves the position of the new x-axis towards the z-axis leaving the direction of the new y-axis fixed. We can follow this with another rotation about the fixed x-axis. The rotation generator formula allows us to find the required rotation matrices for each step and multiplying these in the correct order gives the equivalent rotation. The generator G(a,b) allows us to define a vector function Γ(a,b) which produces a set of matrix coefficients N, P and G for the plane rotations. These are saved in the matrix X and multiplied by the trig functions to give the R_k(θ). The order of the plane rotations in the formula for the equivalent rotation is from right to left since they act on vectors and matrices on their right side.

The matrix coefficients don't just depend on the unit vectors that define them. As this example shows the plane of rotation can depend on a previous rotation so x'=x'(θ₀).