>It's really the same method in different notation. You take the cross product and separate into parallel and perpendicular components, and then you reach the epsilon^2 equation, and it's the same from there.
Bivectors and cross-product aren't just different notation for the same thing if that's what you meant. They're distinct (but very much so related) mathematical structures. For one thing, one's associative while cross products break associativity.
As far as GA sharing a lot with the more comment vector algebra/calc methods. Personally, I'm happy that GA has an attitude of "if it's not broke, don't fix it". It also means there's really not a lot of time lost in the transition due to the compatibility. Hell it's even backwards compatible in the sense that you can still easily retrieve your axial vectors the cross product gave you if you so wish (which cleared up instantly what the exterior algebra folks were doing with their hodge star business when I decided I wanted to explore that perspective later on).
From my view, it goes both ways: geometric algebra/calculus is a more transparent version of the standard approach and the translation back to it is also a relatively small delta to pick up.
Either way of going about what is in essence the same material entails becoming familiar with multivectors, the wedge product, and multilinear algebra, whether you do it through geometric algebra or the standard approach.
For me, the substantial thing geometric algebra gave me so far was a newfound appreciation of the seemingly disparate systems: tensors, differential forms, matrix algebra, and also a newfound appreciation of stuff like determinants, conjugate elements in group theory, lie groups and lie algebras, etc., because it helps clarify the relationship between them, and as another user here said, you can get propelled up into some pretty advanced stuff later on (said user mentions the Atiyah-Singer Index Theorem and Hodge theory, but caveat: I've only recently started tacking a crack at the latter. I will say that, OTOH, it's pretty nice to be able to see something like the wiki on Clifford Analysis and realize its familiar territory from geometric calculus).
Grassmann algebra is a very important part of it, in fact you can reconstruct it in geometric algebra. More generally though, this algebra would be known as Clifford algebra.
>Geometric algebra is, as the article points out, a more powerful version of the usual vector notation
That's not just a gross oversimplification, this is also flat out wrong if what you meant was that it only has vectors. It has more general objects called multivectors through pretty much the same process you get one, two, etc. forms from the wedge product.
In fact, both GA and differential forms build from the exterior algebra, and you can go from the former to the latter through geometric calculus (one key difference e.g. would be the method of reciprocal bases to compute inner products with non-orthonormal bases, rather than explicitly working out a basis and then its dual). So I'm confused about your remark regarding its alleged deficiency vs. differential forms if you pretty much reconstruct it within the GA/GC system (especially regarding working basis-free).
With regards to tensor notation in terms of calculations, if you mean all that index gymnastics, well GC still openly provides that way of computing things out from what you're used to.
What I like about geometric algebra/geometric calculus is precisely the way in which it's nothing new: it's putting everything people use in one system by clarifying the connections between these seemingly disparate systems. Even lie groups/lie algebras can be constructed rather efficiently in the algebra.
Another appealing feature of GA is its ability to make pretty transparent an old theorem from Cartan and Dieudonne that says you can view geometric transformations like rotations, and even translations (in projective geometry) as compositions of reflections.
There's other appealing features like this in terms of classifying and relating different geometries together that harken back to the Erlangen program, but my point is even in terms of concrete calculations, it's not quite right to say it's just a "more powerful version of the usual vector notation" as it includes more general objects than vectors, and still includes a lot of very similar ways of doing calculations (almost a kind of "backwards-compatibility?") you're used to with tensor index calculations, just with the added bonus of making the transition to the tensors used from vector calculus seamless, alongside other added relation to other systems made more transparent.
Bivectors and cross-product aren't just different notation for the same thing if that's what you meant. They're distinct (but very much so related) mathematical structures. For one thing, one's associative while cross products break associativity.
As far as GA sharing a lot with the more comment vector algebra/calc methods. Personally, I'm happy that GA has an attitude of "if it's not broke, don't fix it". It also means there's really not a lot of time lost in the transition due to the compatibility. Hell it's even backwards compatible in the sense that you can still easily retrieve your axial vectors the cross product gave you if you so wish (which cleared up instantly what the exterior algebra folks were doing with their hodge star business when I decided I wanted to explore that perspective later on).