Representation of angular momentum as bivector

Question

I read about derivation of angular velocity and angular momentum here - https://peeterjoot.com/archives/geometric-algebra/angular_acc.pdf

Peeter Joot wrote: "writing the angular momentum as ..." (see on the image below)

I have a question about this formulation. How does author know that $r \land p$ is angular momentum? It's not obvious for me.

Thanks.

Answer 1

In the context of geometric algebra it's usually considered a definition. But, for any vectors $v, w$ and $I$ the right-handed pseudoscalar $$ v\wedge w = (v\times w)I $$ where $\times$ is the cross product. This can be confirmed in components, or if you know that $v\times w$ is the Hodge star of $v\wedge w$ you can confirm that $$ \star X = \widetilde XI $$ for any multivector $X$ and where $\widetilde X$ is the reversal.