and vector - vector projection and rejection
This graph shows the light blue x.a plane of all vectors which have the same violet projection vector
(x.a)/a onto a and the green line defined by the outer product x^a. The intersection of the x.a plane and
the x^a line uniquely determines the vector x. This procedure is the geometric meaning of calculating
the inverse the vector a with respect to geometric multiplication, i.e. geoemtrically dividing
by a vector a:
xa = x.a + x^a,
(xa)/a = (x.a)/a + (x^a)/a = xa/a = x.
The light green bivector r^a = [(x^a)/a]^a = ra and the green bivector x^a are equal:
ra = r^a = x^a,
since the have the same area (scalar magnitude) and orientation. ra = r^a, because r is orthogonal to a. The vector r = ra/a = (x^a)/a is also called the "rejection" of x from a. In geometric algebra the projection and rejection formulas with respect to a bivector have the same form.
You can change the origin, the blue vector a, and the position of the straight green x^a line by interactively dragging the bright red points with the mouse.
The basic idea for this applet stems from the "DEMOinvertible" in Dorst, Mann and Bouma's geometric algebra MATLAB tutorial GABLE.
Soli Deo Gloria. Created with Cinderella by E. Hitzer (Fukui).