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Hmmm.... if I understand your post correctly, it seems like you think the geometric product should be canceled ("cancel" as in cancel culture). And it should b

by rhelz 3y ago

Hmmm.... if I understand your post correctly, it seems like you think the geometric product should be canceled ("cancel" as in cancel culture). And it should be canceled because it's really weird to be adding a scalar to a vector. Let me try to emphasize your case as much as possible: The geometric product of two vectors, a and b is; a * b = a . b + a ^ b I.e. an "inner' or "interior" product, plus an "outer" or "exterior" product. What kind of self-respecting multiplication consists of a sum of two other kinds of multiplications? And why the heck would you even want to add them? Do I have you correct? As a point of departure, let me quote you here: // The general geometric product between two mixed-grade multivectors follows by writing them all out as sums of products of vectors like the above, then cancelling everything out according to xx=1 and xy=−yx for all choices of x=y // Yup. But you do need quantities who have a multiplication table like x and y above, in order to develop the exterior product. And you say you like the exterior product. So, the existence of quantities which multiply like xx=1 and xy=-yx isn't objectionable to you, per se, right? But, as you also say above, the geometric product is just polynomial multiplication with expressions using quantities like x and y with those multiplication tables. E.g., suppose we are multiplying a and b, which are 2D vectors expressed in the basis of x and y: (ax * x + ay * y) * (bx * x + by * y) = (ax*bx)*(x*x) + (ax*by)*(x*y) + (ay*bx)*(y*x) + (ay*by)*(y*y) = (ax*bx + ay*by) + (ax*by-bx*ay)*xy = (a . b) + (a ^ b) So the reason why it's not weird that the geometric product is the sum of two other products, is that polynomial multiplication is not weird. And that's the same reason why it's legit to add a scalar to a bivector: because polynomial multiplication is legit. When you are multiplying out 2d vectors, the number of terms is reasonable, but 3d and 4d vectors it starts to get to be very tedious to do by hand. The solution? Well, we use (a . b) as a shorthand for some of the terms, and (a^b) as a short hand for the rest of the terms. Basically, it's just naming/bundling together terms from a polynomial multiplication in order to treat them as a single entity, instead of a bunch of entities. Its conceptually more tractable that way, and its way easier on your hand when you are writing them out manually. It's really not as bizarre as it looks. I really wish that this view of geometric algebra were front and center in presentations of it. It does seem really weird to come upon a*b = a.b + a^b cold, without knowing what the heck a wedge product is and why we have two different kinds of products added together. Hope that helps, sorry if I didn't really get your objections or address all your points.

ajkjk 3y ago
Thanks for reading! Yeah I have no problem with the exterior product which obeys x ∧ y=-y ∧ x and x ∧ x = 0. But I see no reason to use the same product for xx=1. Exterior algebra does just fine with its two separate products, and combining the two into the same product seems to me to go poorly pretty much everywhere it's done. The two operations are just not the same and when you combine them the rules are weird and you have to un-combine them to actually do anything useful outside of a couple of specific examples. (Also I'll point out that ab = a·b + a∧b is not generally true for multivectors; it only works for vectors. For arbitrary multivectors you end up with a bunch of "mixed" terms, for instance (xy)(yz) = xz = x∧z which you can't write as either a dot or wedge product.)
- 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.

by rhelz 3y ago

ajkjk 3y ago

Thanks for reading! Yeah I have no problem with the exterior product which obeys x ∧ y=-y ∧ x and x ∧ x = 0. But I see no reason to use the same product for xx=1. Exterior algebra does just fine with its two separate products, and combining the two into the same product seems to me to go poorly pretty much everywhere it's done. The two operations are just not the same and when you combine them the rules are weird and you have to un-combine them to actually do anything useful outside of a couple of specific examples. (Also I'll point out that ab = a·b + a∧b is not generally true for multivectors; it only works for vectors. For arbitrary multivectors you end up with a bunch of "mixed" terms, for instance (xy)(yz) = xz = x∧z which you can't write as either a dot or wedge product.)

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.