What is the inverse of a vector?

> 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.

Doesn't seem self consistent. He defines the ab multiplication as dot product plus "extrusion"/bivector (which seems simpler to call convex combinations of 0,a,b,a+b). Then he says aa is a scalar, presumably because the "extrusion" is 0, but you can't have this identity be 0. Just because it's degenerate does not mean it's 0. And the "extrusion" while not a plane is a line in his definition.

> Substantial transitions in the teaching of existing material take generations.

> In 2020 our basic math/science curriculum and pedagogy in high schools and universities has all been pretty well statically fixed for 50+ years (many parts are unchanged in 200+ years), except in computer science where some of the basic ideas are newer than that, and in graduate-level courses that get closer to the cutting edge.

This is wildly incorrect. Even in the past ~20 years we've seen a sea change in our understanding of science pedagogy. Look up the work of Carl Wieman on active learning, or https://www.pnas.org/content/111/23/8410. Inclusive classroom practices are another thing that's come into fashion in the last ~10 years. The curriculum has also evolved; the most obvious thing to point to is the new emphasis on connections to data science in math/stats courses.

If you're someone who doesn't stay up-to-date on pedagogy, then yes, it takes your retirement to bring about a change. But a lot of people, especially those teaching at small liberal acts colleges, have continually evolving teaching practices. There are entire conferences where people get together to talk about college teaching.

> Even in physics, where a better formalism leads to improved physical intuition and deeper conceptual understanding, a transition is an uphill struggle, because symbolic fluency with geometric algebra takes years of practice.

Do you really believe this? To anyone to recognizes that it's the standard stuff in (a clunkier) disguise, it shouldn't take years.

Appealing to these two frictions does not offer a convincing theory of why GA has not been adopted despite being around for, what, 50+ years? The fact that it's worse than existing notation does.

Have you ever spent a few months trying to solve a wide variety problems using GA as a formalism, or tried teaching it to e.g. undergraduates? If not, you are speculating beyond your experience.

It occurs to me that some of these axioms depend on how many dimensions the space has- in 4 dimensions, a vector would have 4 components, a bivector would have 6, a trivector would have 4, and a quadvector would have 1. And so on, in accordance with Pascal’s triangle.