> What does it even mean to have a “mixed-grade” multivector? The product of two vectors has a scalar part and a bivector part. Why? I interpret the mixed grad

> What does it even mean to have a “mixed-grade” multivector? The product of two vectors has a scalar part and a bivector part. Why? I interpret the mixed grade vectors as weighted sums of parallel and perpendicular parts. The product preserves information by letting the product of grade 1 and grade 1 be in grade 0 + grade 2. It's the same interpretation for products of vectors and blades. you get a multivector which preserves the geometric information in a n-1 and n+1 grade multivector. The product has the interpretation as the information preserving product. So I don't really see what you mean when you say: >Either way, GA’s trick of “removing the vector part, then putting it back” is just… weird, I guess? uv = u(v|| + v_|_) = scalar + bivector every vector is decomposable and so I dont get why this grade splitting is unnatural or uninterpretable to you? It's the reason people are always talking about the even subalgebras of geometric algebras. Besides all the this. The purpose of the geometric product, is that it is an associative product over the embedding 2^n dimensional vector space and you get a closed associative algebra.