Dividing vectors like this isn't a "standard" notation but there's no good reason for it not to be.
>>16386532A notational convention along those lines could definitely be reasonable in some setting, although it might not have all the properties you might want from a division operation over a field. E.g. a vector space with a division operation would not form a field, because even when the notation is well defined*, the result of performing the operation is not another element of the vector space you're working in, but rather just a scalar in the underlying field of scalars. *Moreover, the operation will not always be well defined, since e.g. there may not be a unique scalar whose product with B is closest to A. E.g. if you vector space is not topologically complete, and we we can find a sequence of scalars c_1, c_2, ... whose product c_i B --> A converges to A in the limit but never actually reaches it.
>>16386532>closest to Adefine "closest" in terms of vectorspace axioms.you'll find out that vectorspaces by themselves don't have enough structure to define closest beyond things being exactly equal to each other (i.e. their difference is the 0 vector).you need to at least introduce a norm.
>>16386580>you need to at least introduce a norm.And the relationship between normed spaces (a topological concept) and division operations (an algebraic concept) is actually a pretty important topic in math and physics. There are a number of similar theorems addressing these matters, e.g. Hurwicz theorem tells us that the only real-valued, normed division algebras are the reals, the complex numbers, the quarternions, and the octonions. See https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras)See also Frobenius' Theorem for a related concepthttps://en.wikipedia.org/wiki/Frobenius_theorem_(real_division_algebras)
My initial goal was to use this in [math]\mathbb{R}^n[/math] with the usual Euclidean norm to help motivate the dot product for people encountering the first time. Not original, this is done in the "new math" era text "A Vector Approach to Euclidean Geometry" with different notation. But it would certainly be interesting to explore which normed vector spaces this operation is defined in, and how it behaves in those spaces.>>16386622This is definitely not the sort of division you have in a division algebra, because it has a remainder like integer division does.
>This is definitely not the sort of division you have in a division algebra, because it has a remainder like integer division does.Although it is notable that it's the scalar part of quaternion division.
>>16386707One thing I've noticed playing around with different norms on [math]\mathbb{R}^2[/math] is that the notion of "orthogonal" that one might construct from the norm ([math]\forall \lambda, |\vec{A}| \leq |\vec{A} + \lambda \vec{B}|[/math]) isn't in general a symmetric relation.https://www.desmos.com/calculator/ejezqii7vm
>>16387865norm isn't enough to define orthogonalityyou need an inner product for thattrying to overload words with too many meanings confuses ideas.invent a new word, maybe like "normthogonal"
>>16387951>>16387865>norm isn't enough to define orthogonalityNo, but there is a related notion that is defined on normed spaces called Birkhoff Orthogonality.
>>16387967>[math]\|x + \lambda y\| \ge \|y\|[/math]surely that's supposed to say[math]\|x + \lambda y\| \ge \|x\|[/math]
>>16387971apparently so
>>16387971No, and don't call me Shirley.
>>16386580>Define closest in terms of vector space axiomsWhy are you asking for an impossible task? Nobody does that.
