I'm self-studying Grassmann algebra and I find it hard to visualize the regressive product, specifically the magnitude and orientation of the outcome.
My main soure is John Brown's Grassman Algebra which introduces the following axiom with little explanation as to why this should be the case: $$ (A \wedge C)\vee (B\wedge C)=(A\wedge B\wedge C)\vee C \tag{1}\label{1} $$
Now later in the book the author introduces another axiom which captures the dual nature of this product but unfortunatelly requires the space to have a metric: $$ \overline{A \vee B} = \overline{A} \wedge \overline{B} \tag{2}\label{2} $$
This could also be the definition or we could also define it using the interior product...
My questions are:
Is there an intuitive and/or geometric justification for $\eqref{1}$?
How could I prove that in fact $\eqref{1}$ and $\eqref{2}$ are eqvivalent (as a definition of the regressive product)?