Comment by elengyel

My position is that the geometric product and antiproduct are good for one thing, performing transformations with sandwich products q ⟑ p ⟑ q̃ or q ⟇ p ⟇ q̰ and composing those transformations. Literally everything else (join, meet, contraction, expansion, projection, inner product, norm, ...) can and should be done in the exterior algebra without any geometric products.