What are units of the geometric product of two vectors?
Edit: didn't read clearly. Better question is why would we want to have a scalar with units of square meters in this case? Shouldn't 0-grade objects all have the same units?
> Shouldn't 0-grade objects all have the same units?
No, and this is a really important point. Scalars can have any units. For example, you can have a scalar with units of time, or mass, and you can take the dot product of a force vector and a displacement vector to get a scalar with units of [force]·[distance] = [work].
If you instead form the wedge product of those vectors, you get a bivector with the same units: [force]·[distance] = [torque] (note that the units of [work], [torque], and [energy] are all the same).
If you take the geometric product of a force vector and a displacement vector, the result is the sum of a scalar and a bivector, both with the same units of [force]·[distance].
> If you take the geometric product of a force vector and a displacement vector, the result is the sum of a scalar and a bivector, both with the same units of [force]·[distance].
This part is where I have problem with GA: what the hell is the physical[0] interpretation of such a sum? For example, a four-vector <p_x,p_y,p_z,c·E_k> (momentum and kinetic energy) can be thought of as kinetic energy being the temporal component of (4-)momentum, but no similar interpretation seems viable to combine work and torque into a logically unified quantity.
0: I'm not sure if this is the right word - the interpretation as single unified value with no special-case treatment of its components might be another, equally not-quite-right way of putting it.
The physical interpretation depends on context, but the concrete arithmetic is the same whether you write it using GA language or some other mathematical formalism (matrices, Gibbs–Heaviside style vectors, differential forms, tensors, complex numbers, ...), so if you have a problem with it your problem is with the physics.
Like with any expressive language (e.g. English, C++, or matrix algebra), GA makes it possible to state a wide variety of nonsensical things. But that’s not the fault of the language.
Where GA really shines IMO is in the ability to divide by vectors, something that is extremely useful but gets super cumbersome using other mathematical languages.
The biggest “problem” I have with GA is that it takes a lot of practice to get familiarity with all of the powerful stuff it can do. There are pages and pages of (extremely powerful and useful) short identities which are impossible to memorize by just looking at them, and can only be learned as far as I can tell through years of hard-won experience. I feel like I still only really have a handle on the most basic stuff.
I find regularly myself working on some complicated coordinate-based calculation for 3 pages of scratch paper full of mistakes and wrong turns, only to experience déjà vu, re-express the original thing in GA language, and end up with a clear and geometrically interpretable 4 lines of simple algebraic identities which solve the problem. But I’m not always sure if I’d be able to figure out which 4 lines to use right off the bat, without first going through the coordinate slog.
Isn't that just what distinguishing scalar/vector/bivector/trivector/etc is for? I don't have problems with coordinates because I don't bother with coordinate-based calculations in the first place. What I don't get is how adding two multivectors of different [ranks? eg scalar + bivector] is supposed to simplify anything.
> > > the geometric product [...] is the sum of a scalar and a bivector
> any expressive language [...] makes it possible to state a wide variety of nonsensical things.
Yes, but if everything the language makes it possible to state is nonsensical, then what's the point?
I assume there's some sort of point to geometric algebra, but I have yet to encounter any convincing explanation of what that point is, and why I shouldn't keep my dot and wedge products properly separated.
I just said that for many single calculations, GA has saved me literally hours of headaches, in addition to making the result dramatically clearer and more insightful.
Nobody is going to force you to try something you don’t want to try. I don’t think I’ll be able to convince you.
> I just said that for many single calculations, GA has saved me literally hours of headaches, in addition to making the result dramatically clearer and more insightful.
Er, right; I was[0] asking for a example of a such a dramatically clearer and more insightful result, relative to just using normal dot and wedge products. Using vectors at all saves hours of headaches if all it's being compared to is coordinate-based calculations.
0: Well, the comment before that one was asking about the interpretation of the geometric product.
It’s kind of the same, actually. You can treat the individual entries as “components” in a 2^d-vector with special multiplicative structure, which essentially transforms in specific ways. These axioms of GA lay that out more clearly: https://arxiv.org/pdf/1205.5935.pdf (specifically axiom 6)
As I said, the units of the geometric product of two vectors (which if you like can be broken into the sum of a scalar part + a bivector part, each with the same units) will be the product of the units of each vector.
If you multiply two vectors with units of meters, you get a scalar + a bivector, both with units of square meters.
If you divide two vectors with units of meters, you get a scalar + a bivector which are both dimensionless.
Etc.