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Perhaps 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 d

• ajkjk 3y ago
  Well 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.
  • rhelz 3y ago
    // Its not just // Come on man, polynomials don't have to be only over complex numbers and their subsets...its algebra, not algebra ;-) In fact, it's not even just polynomial multiplication, it's just plain multiplication. The geometric product of two scalars is the product of the scalars. The geometric product of a scalar and a vector is the product of the scalar and the vector. Geometric multiplication is called "multiplication" because its multiplication. Its just a coherent superset of the kinds of multiplication you do on all other numbers/elements of the field/whatever. // doesn't have any geometric interpretation // Still don't understand why you don't think it does. I mean, the dot product? The exterior product? Are they not full of geometric meaning? Is it just the fact that they are added together that you don't like? Hmmm....I don't know whether this is helpful or not, but try this on for size, you are a programmer right? Most scripting languages have a data structure like a hash-map, where you can associate a key and a value, ya? Well....a multi vector is a hash map. Really. The basis vectors, basis bivectors, etc, are the keys. The coefficients on them are the values. Come to think of it, this way of viewing it should make the grade operator make sense as well: its just like the [] operator on a hash map: if a = ax*x +ay*y, then a[x] = ax, and a[y] = ay. (Well, not just like the [] operator; the grade operator returns a new hash map of all of the key-values pairs for which the keys are of the given grade. But you get the idea). The geometric product is then a function which returns a hash map. If you pass in two vectors, the returned hash map will map the key 1 to the dot product, and map the key xy to the exterior product. The function is implemented as a call to the "multiply polynomials" subroutine. This way of looking at it has been a great intuition pump for me, as well as providing for a bridge between my knowledge of programming and my knowledge of geometry. Before I thought of multi vectors this way, I would find myself thinking things like "man, if I could just bundle up these values in a hash map and then feed it to a function..." Now I can!! The vectors <=> polynomials <=> hash map isomorphism allowed me to bring my full brain to bear on geometric problems. The facility I had with manipulating polynomials and the facility I had with hash maps could be brought together in a unified methodology. The utility of the geometric product then is kind of like the utility of operator overloading: imagine you had to have a different operator to multiply ints, reals, and complex numbers? Imagine if you wanted to add them together, you'd need a separate kind of operator for real + int, real + real, real + complex, etc etc. Madness. The push to express everything as geometric multiplication is the same push as to write general purpose, reusable functions when programming. helpful at all?