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What is the inverse of a vector?

mattferraro.dev

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Re: What is the inverse of a vector?

#111

Post by civilized »

Grustaf wrote: post #98
fouronnes3 wrote: post #77

As a programmer it seems to me that the number one problem of math notation is that it's weakly typed. There's abuse and reuse of notation everywhere, which makes learning it needlessly difficult. I want a strongly typed fork of math notation. 90% of existing math notation would just be laughed at if it had to go through code review.

That's a very common criticism, but I don't think mathematics would work if you insisted on being 100% explicit all the time. Clear and short notation, that is just unambiguous enough, is a very important factor, without it books wouldn't just be much longer, I'm not sure we'd even be able to understand it.

This was what Bourbaki's Elements and Whitehead/Russell's Principia Mathematica were about. These books are admired and influential but very few people actually read them. As you might expect, they're too long. They're for giving a different perspective to people who have already achieved the highest levels of sophistication in math.

Re: What is the inverse of a vector?

#112

Post by Rayhem »

SantiagoQ wrote: post #47

Semantically, the inverse of a vector is something that has no magnitude nor direction. I wonder what would that look like (mathematically), and what surfaces or fields would it create?

"A vector is a thing with both magnitude and direction" isn't really a good definition. Cars have both -- an SUV is larger than a sedan, establishing a magnitude, and they obviously point in a direction -- but I don't think anyone would mistake them for pointy arrow vectors. If you use the more rigorous definition that a vector is an element of a space that obeys the vector space axioms it becomes easier to invert a vector "semantically" (a thing that doesn't obey the axioms) but quite a bit less useful. Cats don't obey the axioms, nor do punctuation marks.

Re: What is the inverse of a vector?

#114

Post by spacedcowboy »

azalemeth wrote: post #46
vanderZwan wrote: post #20

Earlier quoted context omitted.

> It seems that this is written out of a perspective of some missing knowledge. Well, the author talks about what "we learned in school ", not university , so that checks out but only because you two have different audiences in mind.

Indeed. I learnt about vector dot and cross products, basic linear algebra (including diagonalisation, simple Markov chains and similar), partial differentiation, grad div and curl, the volume of a parallel piped and "all that Jazz" in high school, as a 17-18 year old. I learnt about the divergence theorem, Stokes's theorem, multivariate integration, integrating factors and higher order ODEs and simple PDEs, Fourier…

I'll second this - my experience is pretty much the same (though I did some basic Fourier transforms at school), studying Physics at college. I did 3 Maths A levels along with Physics and Chemistry (and "General Studies" :) however, so I too may be atypical.

Re: What is the inverse of a vector?

#115

Post by necovek »

I love the way this was presented (others have pointed out flaws with the article already).

I'd appreciate a post from Matt Ferraro on how this is built. Bonus points for including nice syntax-highlighted code "widget" for a cross between maths/programming.

Re: What is the inverse of a vector?

#116

Post by spacedcowboy »

chriswarbo wrote: post #36
spekcular wrote: post #7

The article begins: >In this post we will re-invent a form of math that is far superior to the one you learned in school. The ideas herein are nothing short of revolutionary. and concludes: > I firmly believe that in 100 years, Geometric Algebra will be the dominant way of introducing students to mathematical physics. In the same way that Newton's notation for Calculus is no longer the dominant one, or that Maxwell's…

> Geometric algebra is, as the article points out, a more powerful version of the usual vector notation > the stuff that's already taught is better These two statements seem contradictory. > But it is deficient in various ways when compared to tensor notation (for calculations) and differential forms (e.g. if you want to work basis-free) The author made no claims about tensor notation or differential forms; perhaps t…

> I never encountered tensors or differential forms.

Your UK undergraduate physics must have been a bit different to mine. About a third of my physics course was taught by the maths dept, and tensors/algebras were very much a part of that.

I recall, after freshers week, the dean getting everyone together. He said two things:

- Hopefully you all had a great fresher's week, now it's down to business, and

- Make sure you have fun at college.

He also had a projection on the overhead saying "If you can't blind them with science, baffle them with bullshit". I'm reasonably certain the second statement above was the latter, because...

He then casually mentioned a "maths refresher" 2 week course that all freshers had to take before "the real stuff" started. That "maths refresher" was the entire Further Maths 'A' level syllabus. In two weeks. Those of us who had done Further Maths at school were fine. Those that hadn't were shell-shocked.

Re: What is the inverse of a vector?

#117

Post by aktenlage »

foxhop wrote: post #62

Here is something similar I wrote a long while back as notes for my future selfs: https://www.foxhop.net/vector-math-for-video-games

In what way would that be similar? The OP performs a theoretical derivation of geometrical algebra, you wrote a well documented python class with the most basic operations on 2d vectors.

Re: What is the inverse of a vector?

#118

Post by Cybiote »

chriswarbo wrote: post #110
Cybiote wrote: post #86

Earlier quoted context omitted.

spekcular is correct. One could draw the following graph very roughly capturing how these concepts developed: Grassman’s exterior algebra -> Differential forms Cartan’s Exterior Derivatives -> Differential forms Quaternions -> Clifford Algebra exterior algebra -> Clifford Algebra -> geometric algebra Differential forms are a particular kind of tensor and tensors can be defined in terms of multilinear maps. As spekcul…

> As spekcular says, the standard curriculum covers differential forms, tensors and vectors... making geometric algebra a relatively small delta to pick up. Could you be a bit more specific about which "standard curriculum"/"standard approach" you're talking about? For example, in my formal education (high school; masters with physics major, comp. sci. minor; 4 years of a comp. sci. PhD (abandoned)), I did not encoun…

By standard approach I mean the typical material covered for someone studying vector calculus properly. This will be stuff like differential forms and the basics of tensors, manifolds and multilinear maps at the undergrad level. Differential geometry and cohomology are examples of courses which build on them.

I agree with you that pseudovectors, cross products and vector calculus are a terribly adhoc way to teach this stuff but a course covering linear algebra with differential forms elegantly unifies, corrects and generalizes them. Standard is also in contrast to the geometric algebra/calculus alternate path.

Re: What is the inverse of a vector?

#119

Post by Cybiote »

LotusFunctor wrote: post #97
Cybiote wrote: post #86

Earlier quoted context omitted.

spekcular is correct. One could draw the following graph very roughly capturing how these concepts developed: Grassman’s exterior algebra -> Differential forms Cartan’s Exterior Derivatives -> Differential forms Quaternions -> Clifford Algebra exterior algebra -> Clifford Algebra -> geometric algebra Differential forms are a particular kind of tensor and tensors can be defined in terms of multilinear maps. As spekcul…

From my view, it goes both ways: geometric algebra/calculus is a more transparent version of the standard approach and the translation back to it is also a relatively small delta to pick up. Either way of going about what is in essence the same material entails becoming familiar with multivectors, the wedge product, and multilinear algebra, whether you do it through geometric algebra or the standard approach.

That makes sense but my argument is since further material (some examples which I listed) assumes and builds upon the standard approach, you'll likely be better off taking that path.

Re: What is the inverse of a vector?

#120

Post by adrian_b »

da39a3ee wrote: post #94

> A scalar is a point on a number line. This is going to confuse readers. A point on a number line is a 1D vector; in other words it is a unit vector pointing along that number line, multiplied by something which scales its length. It’s the latter dimensionless and directionless quantity that’s the scalar.

Nitpicking: a more correct view is that the set of points on a straight line is an affine space, so the points are neither scalars nor vectors, but elements of an affine space.

The set of translations of the straight line is a vector space a.k.a. a linear space.

So the vectors are the classes of equivalences of the differences between 2 points on the straight line (i.e. the differences between 2 pairs of points, where the distances are the same, are equivalent and they determine the same vector).

While the vectors are classes of equivalence of the differences between 2 points, the scalars are classes of equivalence of the quotients of 2 (collinear) vectors, i.e. a scalar is the ratio between the signed magnitudes of 2 collinear vectors.

If you choose a point on the straight line as the origin, you can choose as a representative of each class of equivalence that corresponds to a vector, the vector corresponding to the origin point together with another point. This gives a bijective mapping between vectors and those second points.

If now you also choose a vector as being the unit vector, which will correspond with a second point besides the origin point, together with the origin point, then you can choose as a representative for each class of equivalence corresponding to a scalar the ratio between a vector and the unit vector, which will correspond to a third point, besides the origin and the point corresponding to the unit vector. So you obtain a bijective mapping between scalars and those third points.

Because on a straight line there are bijective mappings between points, vectors and scalars (after choosing 1 origin point and a 2nd point as the extremity of a unit vector), they can be used interchangeably in most contexts, but it would be good to remember that all 3 are in fact different mathematical entities.

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