[jacobolus]

> Suppose I want to teach first-semester mechanics

If you need to teach undergraduate mechanics, I highly recommend you at least read some of Hestenes’ New Foundations for Classical Mechanics http://geocalc.clas.asu.edu/html/NFCM.html

> without any distinctive advantages

The most basic distinctive advantage is that you can invert vectors (which is incredibly useful!!) without needing to pretend that vectors are matrices, complex numbers, or some other kind of object.

GA takes most of the advantages of complex numbers vs. R² for representing plane geometry, but extends them to arbitrary dimension, and extends them further (when using complex numbers for plane geometry you end up representing vector–vector products via the obscure z̄w product involving complex conjugation, and it is easy to get confused about the difference between a vector vs. a scalar+bivector).

But there are a wide variety of other powerful (and geometrically interpretable) algebraic identities which can be applied to vectors, blades, and multivectors, ranging from awkward to impossible to express using the language of differential forms, Gibbs-style vectors, etc. Physicists often end up resorting to tedious coordinate-by-coordinate calculations for stuff that would end up being an easy vector expression in GA. Learning these identities and how to apply them takes years and a lot of practice solving problems using GA.

My own experience for the first few years of knowing that GA existed but not being too fluent with it was that I would work some problem (mostly 2–3 dimensional geometry problems) out in coordinates, spending like 2 pages of scratch paper for the opaque intermediate calculations, with high chance for mistakes, then eventually find that most of the ugly bits along the way canceled and yielded a nice result. Then I would think a bit more about the problem, skim through a list of GA identities, and find I could have shortened that 2 pages of work to 3 lines, each of which had an obvious geometric interpretation.

[spekcular]

Can you give an example of a problem that might appear in an undergraduate physics or math course, whose solution is lengthy and tedious by "usual methods" but dramatically simplified by the use of GA?

I have seen examples proposed before and been distinctly unimpressed. Any serious simplifications in solutions are usually due to some notation-agnostic insight.

[jacobolus]

The nicest written-up example I know is https://www.shapeoperator.com/2016/12/12/sunset-geometry/

* * *

As a relatively recent personal example I spent a few months (in bits and pieces) working out a bunch of metrical spherical geometry for myself without reference to past work, with points represented as displacement vectors to stereographically projected points at https://observablehq.com/@jrus/planisphere with the eventual goal of implementing computational geometry / cartography code using that as a canonical representation, which I think is superior to representations used currently in practical software.

The same spherical relationships are comparable (some things slightly slightly easier, some slightly trickier) to represent as displacement vectors in an embedded sphere. But there again relationships are clearer to express in GA terms.

Most of the material there is stated without proof (maybe eventually full proofs should be included), but several of the identities there I worked out very tediously with pages of scratch work in coordinates, then realized afterward the same results could be arrived at with only a few lines of GA.

Only a bit of the material is truly novel (after doing the work myself, I hunted around for sources and found some of the same formulas worked out previously using classical spherical trigonometry 200+ years ago), and e.g. some very similar material where the stereographically projected points are represented as complex numbers can be found at http://fer3.com/arc/img/110279.applications%20of%20complex%2...

In theory most of the rest could be also worked out using complex numbers or matrices, but (a) some ideas end up awkward and unidiomatic there so you would never think to do it, so that many identities that are slightly obscure in GA are almost unheard of written in other formalisms, (b) the algebraic manipulation is at least 2–3x more cumbersome.

[spekcular]

That is a strange and unconvincing article. In outline, it goes:

1) Look at this slick solution using geometric algebra.

2) Look at how ugly the trigonometric solution is. GA is so great!

3) And, by the way, one can mechanically translate the GA solution into the usual vector notation.

Point 3 is even a bit understated: GA concepts are really only used for a few lines under "Solving for the Earth's radius." Once you hit the equation for epilson^2, it's just standard algebra and trig.

Anyway, the relevant comparison is not between trig and the GA solution; it's between GA and the usual vector language. 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.

Also, I think the author is far too hard on the trigonometric solution. The vector solution is somewhat clever, and it for any given problem that gets placed in front of me, it's not obvious a slick solution exists. On the other hand, the philosophy of trigonometry is that given a completely determined problem about triangles, you can just trig-bash mechanically to get an answer (and here you can even before starting that small-angle approximations will make life easier, so trig is even more attractive). It's really not that bad here. Especially, the comment about it being tricky because one must find a "non-trivial relationship between the four angles" is puzzling. Anyone who's spent time with geometry problems like this knows that the first step is to angle-chase and write in all the value and relations, from which this falls out immediately (and again, totally mechanically). Then you just turn the algebra crank and win.

[I don't have time to think about the spherical geometry stuff. Sorry!]

[LotusFunctor]

>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).