The rotation of an orange vector x into the violet vector RxR~ is described with the rotor
R = exp(I Phi/2), and its reverse R~ = exp(-I Phi/2).
The unit bivector of the (light blue) plane of rotation is I. The bivector angle I Phi is indicated by the light blue hexagon. The (blue) angular vector of rotation in the direction of the axis is (I Phi).I3, i.e. the dual of I Phi.
In geometric algebra the same rotor formula for rotations applies to any multivector element. In the graph you see the example of rotating the yellow B-plane (indicated by the bivector B) which contains the vector x into the violet plane RBR~ which contains the vector RxR~. For better visibility the nodelines of the planes B and RBR~ with the light blue bivector angle I Phi are indicated in orange and violet colors.
By dragging the bright red points with the mouse you can interactively vary the angle Phi, the vector x, the dark green perpendicular component of x and the magnitude |B| of B. If shapes get weird, etc. just use the refresh button of your browser.
The basic idea for this applet stems from the "DEMOrotor" in Dorst, Mann and Bouma's geometric algebra MATLAB tutorial GABLE.
Soli Deo Gloria. Created with Cinderella by E. Hitzer (Fukui).