Your first point got me curious, though. I found E&M in differential forms quite satisfying from an intellectual point of view (although of little practical benefit to the problems I was interested in compared to the classical approach, which I was plenty familiar with already). What advantages do you see for GA vs. differential forms in this context? This might make for a few interesting seminars with our math folks...
One advantage I see in the GA treatment of E&M is in radiation. If one takes the Faraday bivector for a stationary charge (ie. the Coulumb field) and then just 'rotates' it into a moving frame using the GA rotors you can find the standard radiation fields of E&M quite easily. This is cool because it's not only a neat proceedure mathematically, but its computationally quite efficient, say if you need to find the fields produced by N moving charged particles.
I'm sure there are more examples of things like this but it's been a while since I last looked at E&M in GA so none are coming to mind right now.
To get radiation fields, you need Maxwell's equations and boundary conditions. When you have those, what is the use of GA?
1) Start with the Coulomb potential A(r) = qγ₀/(4πr)
2) make it covariant by replacing r -> r -> X⋅γ₀ where X = r(γ₀ + γᵣ). This is a trivial transformation in the static frame as one can easily show X⋅γ₀ == r, but it's important as it encodes the causal information you astutely pointed out is vital to obtaining radiation solutions.
This gives A(r) = qγ₀/(4π X⋅γ₀)
3) Now, one can 'rotate' this to a frame with (not necessarily constant) velocity v via A'(r) = R A(R̃ x R) R̃ ie. evaluate the rotated A at inverse rotate spacetime position. Then
A'(r) = q v/(4π X⋅v)
This is the Liénard–Wiechert potential (this can also be obtained by solving ∇²A = J using the retarded Green's function G(r, t) = δ(|r|-t)/(4π|r|).
4) Now once can get the radiation fields by taking the derivative of this and doing some algebra:
F = ∇A = (q/4π) (X ∧ v + ½ X(v̇ ∧ v)X)/(X⋅v)³
The term proportional to X ∧ v is just the Coulomb field, but the term with the X(v̇ ∧ v)X is the radiation field which one can show propagates to infinity.
This is unfortunately much less elegant and straightforward than I let on initially so I apologize.
I still think this has advantages over standard treatments but your milage may vary.
Preparing for the additional requirements of the lectures of the coming semester is whar the semester break is for.
Especially important in high school is to dive into not only the “complex numbers” but also the “split complex numbers”.
I would argue that E&M and anything involving rotations (or Lorenzt boosts!) are two great examples where GA is miles ahead of anything other than differential forms but still has modest advantages of differential forms as well.
The main advantage I see in geometric algebra is that its a unifying framework. The same langauge used to understand E&M in GA is also useful in quantum mechanics, general relativity, complex analysis, linear algebra, etc. I'd argue that physicists don't mind this mish-mash of notations because they've already put in the work to learn them all separtely but I think there are real advantages (especially pedagogical) to unifying them into one framework.