Ah, another geometric algebra evangelist? I can't figure out if GA actually adds anything substantial, or if it merely lets us write some equations in a more succinct fashion. But it certainly looks cool. As to vectors, obviously they have inverses, additive inverses. Since vectors don't have multiplication, there is no multiplicative inverse, but if you define new operations on them, well then that operation can hav…
Basically the extra stuff that is usually skipped in first year linear algebra courses are the symmetric and asymmetric (often called exterior) products. These form algebras, of course. The exterior product, or wedge product, has a natural interpretation in terms of signed areas (or volumes) and from this you get the determinant as a volume form.
These are the natural generalizations of dot products (inner products) and wedge products (exterior products).
You can take a vector and associate it with a 1 form (asymmetric algebra or exterior algebra), and then multiply two vectors to get a 2 form using the standard wedge product, etc. In dimension 3, the space of 2 forms is dual to the space of 1 forms and so you can "multiply" two vectors to get a third vector. That is all that's going on here.
Actually a good multi-variable calculus class will cover most of this stuff as you need some motivation for Jacobian volume forms used to calculate areas and volumes under change of basis, and dot/wedge products are useful for generalizations of the Gauss divergence theorem and the generalized fundamental theorem that says the integral over a function, f, on the n-1 dimensional boundary of a shape is the differential of the integral of the shape.
Moreover any class on Riemannian geometry will give you all the linear algebra you need as well.
One thing I would caution students with is that by using somewhat non-standard jargon they may not understand how to generalize this stuff to n-dimensions, nor will the connections between, say, determinants and wedge-forms be clear, or dot products and angles be fully understood if only the n=3 cases is emphasized. Only in n=3 can you multiply two vectors to get a vector. But fun fact: in dimension 3k you can multiply two k-forms to get a third k-form (as the space of 2k forms is dual to the space of k forms in n=3k). If you think there is this new thing called "geometric algebra" other than usual tensor products, it may not be obvious how things generalize to n != 3.