Projection and rejection of vector from/on a bivector

The red vector r = (x^B)/B is the rejection of the vector x from the bivector plane B. The violet vector (x.B)/B is the projection of the vector x onto the bivector plane B. The blue vector (x.B) is the scalar product of the vector x with the bivector B. The light blue parallelepiped shows the trivector x^B. Reshaping x^B into the orange-yellow form x^B = r^B = rB shows that dividing with B yields the rejection vector r:

r = rB/B = (r^B)/B = (x^B)/B.

It is instructive to compare the projection (x.B)/B and the rejection (x^B)/B with respect to the bivector B to the projection (x.a)/a and the rejection (x^a)/a with respect to a vector a.

You can change the vector x by interactively dragging the bright red point with the mouse.

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Soli Deo Gloria. Created with Cinderella by E. Hitzer (Fukui).