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|
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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.
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Using orbital period as the time unit for the velocity vector,
The blue parallelogram is twice the area of the red ellipse.
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Specific angular momentum = (2 * area of the ellipse)/period
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When the position vector is at peri or apohelion,
the bivector is a rectangle and its area is simply |r| * |v|.
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A good book about bivectors and other Clifford Algebra concepts is
Clifford Algebra & Spinors by Pertti Lounesto