Antidot Product

Projective GA
1

In Eric Lengyel’s article “Projective Geometric Algebra Done Right” (Projective Geometric Algebra Done Right – Eric Lengyel's Blog) he introduces a new product, the antidot product. He briefly mentions the symmetry between the dot and antidot product with the following quote “The antidot product works with antivectors in the same way that the dot product works with vectors, but it produces an antiscalar quantity instead of a scalar quantity.”

How exactly is this product defined? I referenced Eric’s Cayley table (http://projectivegeometricalgebra.org/projgeomalg.pdf) and can’t understand how the geometric antiproduct (which includes the antidot product) between the vector e4 and itself can result in the negative antiscalar. All the products in the table that resolve to just the antidot products are just as opaque.

Any help is much appreciated.

2

Easier reference is at Dot products - Projective Geometric Algebra

3

Thanks for sharing the link. e4 ⋅ e4 = 0 definitely tripped me up.

I believe the relation between the dot and antidot is similar to that of the wedge and antiwedge. Given two antivectors (n-1 vectors) A and B, their antidot can be defined in the following way.
A ∘ B = D-1(D(A) ⋅ D(B))

If this is the correct relationship then the antidot product between antivectors makes sense. However the question of the dot product between elements of grade greater than one, like the bivectors and trivectors or equivalently the antidot of elements with grade less than n-1 is still not clear to me based on that definition.

5

For example, e23 ⟑ e23 = -1 which is a scalar. This has to be the contribution of dot product only since wedge product of those is zero.