[howling]

> Yes, sometimes you just need the dot product, and sometimes you just need the exterior product. If you are coding, or giving the final form of some formula, you don't have to always put both of them in your code or paper.

In my experience 99% of the time you just want the dot product or the exterior product. Even when you want both it is rare that you want to combine them linearly except in some niche physics/mathematics.

> But neither the dot product nor the wedge product are investable by themselves. Having an investable product on vectors is endlessly useful while you are deriving the formulas.

Do you mean invertible? Why is invertibility is so useful?

[wvlia5]

Yes, invertible like if you have a.x=b, then you can find x=b/a if . is the geometric product.

Why? Well, solving equations sounds somewhat useful, right?

[howling]

First of all it is only invertible for some non-zero elements, especially if `a` is a linear combination of multivectors or we work in PGA that explicitly adds a basis vector of norm 0. Yes sometimes it is useful but that doesn't automatically makes it more fundamental than the inner product and exterior product.