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 n

_{1}and n

_{2}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.