Parallelogram with sides r and v is the bivector r /\ v.

(r /\ v reads "r wedge v")

Magnitude of r /\ v is the area of the parallelogram.

|r /\ v| = |r| * |v| * sin(gamma) = |r X v|


r /\ v spins counterclockwise and v /\ r spins clockwise.

r /\ v = - v /\ r.

Like the cross product, the wedge product is anti-commutative.

r X v defines orbital plane because it's perpendicular to it.

r /\ v defines orbital plane because the bivector lies in it.

As r gets longer, v gets shorter. Parallelogram area is constant.


Using orbital period as the time unit for the velocity vector,

The blue parallelogram is twice the area of the red ellipse.


Specific angular momentum = (2 * area of the ellipse)/period


When the position vector is at peri or apohelion,

the bivector is a rectangle and its area is simply |r| * |v|.


A good book about bivectors and other Clifford Algebra concepts is

Clifford Algebra & Spinors by Pertti Lounesto

Applet showing position and velocity vectors