nD Rotation Generators

  The rotation matrix for rotations in a plane is of the form R = I cos(θ) + G sin(θ) where G plays the role of a generator. This matrix only rotates vectors in the plane properly. One can show that

  R(θ) = I - (I - e^G·θ)P_ab

is the correct matrix for rotating any point in n-space where I is the identity matrix and  P_ab is the projection operator that projects a point onto the ab-plane. Here's a worked example. The vectors n₁ and n₂ are unit vectors normal to the vectors a and b that define the plane of rotation. One can see that the projection of z onto the ab-plane has magnitude 1 and the normal projection has magnitude 2.5. After the rotation the projections for z' are the same. A check shows that the angle of rotation is the desired angle β.

The computed values for the vectors and rotation matrix are shown below.