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Hmmm....so basically, you are saying that you prefer to keep two sets of books, one for the inner product and one for the outer product? Its true that when you
by rhelz 3y ago
Hmmm....so basically, you are saying that you prefer to keep two sets of books, one for the inner product and one for the outer product?
Its true that when you are, say, proving that two expressions are equal, e.g.:
a = b
you might have to "unpack" each of the components, and separately equate them, e.g.:
ax*x = bx*x and
ay*x = by*x so ax = bx and ay = by
etc etc, but frequently, there are a lot of intermediate steps before that, and I've always found it easier to reason about those steps at the higher level of combining the two products into a single product.
I guess I still don't understand why you think the rules are weird: its just polynomial multiplication, what's so weird about that? Yeah, the operations are not the same, but they are come from one operation, viz, multiplying two polynomials.
Sure ab = a.b + a^b is only true if a and b are vectors. But if they are general multi vectors, computing ab is still just a polynomial multiplication. And yeah, not every geometric multiplication can be expressed as (a.b) +(a^b), which is to say that we have not invented a shorthand operator for every possible subset of terms for every possible result of multiplying two polynomials together. Doesn't mean its not useful shorthand tho!
The real big payoff (as another commenter mentioned in another thread) is that the geometric product is invertable. Neither (a.b) or (a^b) is invertible by itself. But their sum is. The fact that geometric multiplication is invertible just makes it soooo much easier to use all your algebra skills when manipulating and deriving expressions.
......
Do you remember the first time you learned about vectors? I do....I had just spent 4 or 5 years learning that you can multiply and divide integers, and integers are just a subset of real numbers, which you can also multiply and divide. And then complex numbers, which are a superset of the real numbers, which you can multiply and divide.
So, naturally, I just assumed that you would be able to multiply and divide vectors, and add scalars to vectors, just like I added integers to real numbers.
But the teacher just said "you can't multiply or divide vectors, MmmKay?" No reason. No explanation. "you can't add vectors to scalars, MmmKay?"
YMMV, but I would have enjoyed the experience much more if I had known that you actually can multiply and divide vectors, and add them to scalars, etc. And that multi vectors are a superset of real and complex numbers, just like they are supersets of integers. The whole thing would have cohered a lot better than just an arbitrary "stop asking those kind of questions" did.
And learning geometric algebra in the 9th grade (which is possible, because you've already learned polynomial multiplication), and therefore, ipso facto, learned the exterior algebra, would have been endlessly useful for the rest of my life. Imagine learning real physics in high school.
- ajkjk 3y ago> its just polynomial multiplication, what's so weird about that? Well that's the point of the article. You can do it, but the point of GA is to make vector algebra and calculus easier, and the geometric product doesn't do it. So stop teaching it to people. It just makes the problem worse: "you CAN multiply vectors, but it's even more confusing, yet we're going to act like it's even less confusing". I agree that it was weird, back in the day, to not be able to multiply vectors... but I would say, yes, you can, it's called the tensor product. Out of all the operations: geometric product, tensor product, wedge product, dot product... the geometric product is the least useful and least intuitive. So stop telling people that it's fundamental and reframing other things in terms of it. Just nix it entirely and everything makes a lot more sense. Re invertibility: Yes, the geometric product is invertible (in positive metrics), but what are you inverting? Inverting something that isn't meaningful just produces something else that isn't meaningful. The cases where it's useful is when the thing you're inverting is a specific operator, like a rotation, and in that case you may as well just invert that operator, not the geometric product in general.
- rhelz 3y agoPerhaps I would understand your point better if you showed the same derivation both using GA and keeping two books for the two separate products? Honestly, I don’t get why the geometric product is unintuitive and not meaningful: it’s just polynomial multiplication, which I have known what it means since 9th grade, and have very strong intuitions about.
- ajkjk 3y agoWell it is not just polynomial multiplication, because polynomial multiplication has xy = yx whereas the geometric product has xy=-yx for orthogonal vectors and xa=ax for parallel vectors. Actually the thing that is "just" polynomial multiplication is the "symmetric" product, defined as a×b = a⊗b + b⊗a and extended to be symmetric on all arguments in larger products. It obeys eg xyxxy = x^3y^2 for all factors without anything cancelling out. The geometric product, on the other hand, is some weird intermediate operation between the symmetric product and the antisymmetric product. But again, the complaint is that the geometric product of two multivectors doesn't have any geometric interpretation. It is just a thing you have to deal with. Whereas in polynomial multiplication, and in exterior algebra, every single intermediate object means something. For instance in a polynomial like abc^2 you can talk about objects like (abc) and (bc^2) and (c^2) and assign meaning to them. In a geometric product like abcc you can't assign any meaning to any of the objects (except for cc = |c|^2) without knowing more about them. So it is uniquely un-meaningful out of the objects you can do vector algebra with.