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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?

#131

Post by mferraro89 »

an1sotropy wrote: post #45
ur-whale wrote: post #15

Geometric algebra (Clifford Algebra) unfortunately came late historically. It's a shame, because the whole theory is a very useful (eg for engineering / applied math) superset of linear algebra. I really wish I had learned this first in my undergrad years, would have made a whole bunch of things way clearer from the get go: differential forms tensor calculus linear algebra etc From zero to geo is a very good video in…

Yes, this. Though Zero to Geo is one of the links at the bottom of the article. It is really a shame that article does not clarify that, btw, what we've just derived is a re-derivation of a thing that has already been expressed and named, by Clifford, and well-characterized: https://en.wikipedia.org/wiki/Geometric_algebra Such a bummer to see very slick but very ahistorical articles.

Hi. I wrote whole section on the history of GA and what happened and why it isn't already the norm, but I chose to remove it because the article is already far too long, and I don't think that my intended audience (engineers, compsci people, university undergrads) would care about the history. Apologies that wasn't what you would have preferred.

Re: What is the inverse of a vector?

#135

Post by 6gvONxR4sf7o »

LotusFunctor wrote: post #35
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 That's not just a gross oversimplification, this is also flat out wrong if what you meant was that it only has vectors. It has more general objects called multivectors through pretty much the same process you get one, two, etc. forms from the wedge product. In fact, both GA and differential forms build from the exte…

> ... by clarifying the connections between these seemingly disparate systems.

This is the big reason I like it. I remember learning bits and pieces of linear algebra whose rules seemed so entirely random (vectors vs pseudo-vectors, curl, quaternions, some spin calculations I can't even remember anymore, etc), that turned out to have more unified geometric interpretations once I learned GA. It definitely slowed me down as a student who hated 'just memorize it' pedagogy.

Re: What is the inverse of a vector?

#136

Post by spekcular »

matt-noonan wrote: post #83
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…

> it is deficient in various ways when compared to [...] differential forms (e.g. if you want to work basis-free) There is nothing basis-dependent in Geometric Algebra. This presentation started from a basis, but then again so do many presentations of differential forms, leading to 2-forms like dx \wedge dy and so on. The actual difference is that Geometric Algebra requires a choice of inner product (actually, you ca…

I worded that in an unclear way. My main beef is that GA is just way clunkier than differential forms, which are clearly the "right" approach if you want to approach the subject from a theoretical perspective. I see no advantage over the usual treatment, and many disadvantages.

Re: What is the inverse of a vector?

#137

Post by mferraro89 »

deepsun wrote: post #24

English/American style of explanation fascinates me. First, they show some algebra formulas and mention dot product and cross product. But then they start introducing a definition of a vector! With images! Why, oh why do you need to waste yours and reader's time to introduce basic definitions, if any reader of the article definitely knows that? If they haven't, they wouldn't be able to read the first paragraph at all…

ha, thank you for your honest feedback. My intended audience is not PhDs or math majors, it is high school physics teachers, practicing engineers, programmers, precocious high school students, etc. Many of these people benefit from some definitions.

I include 3 sentences defining a scalar so that I could introduce the concept of grade.

I include a few sentences defining a vector because just read the comments here and you'll see there are many definitions of vector and I want to specifically call out the one I care about in this post. I am also using a nonstandard, color-based notation throughout the article so it is helpful to take a concept that people already know just to demonstrate my notation. This also lets me introduce the 3D interactive illustrations.

Did you read the rest of the article or were these two definitions so objectionable that you quit?

Re: What is the inverse of a vector?

#140

Post by mferraro89 »

IshKebab wrote: post #28
deepsun wrote: post #24

English/American style of explanation fascinates me. First, they show some algebra formulas and mention dot product and cross product. But then they start introducing a definition of a vector! With images! Why, oh why do you need to waste yours and reader's time to introduce basic definitions, if any reader of the article definitely knows that? If they haven't, they wouldn't be able to read the first paragraph at all…

I definitely agree but there's no way this is "English/American style". It's because people have grand plans of making their article/book accessible to everyone, and they start off explaining e.g. what a vector is, but pretty soon realise the don't want to write an entire vector algebra textbook so they seamlessly give up and jump straight into Stoke's theorem or whatever. I read a Synopsys simulator manual that expl…

Did this article at some point give up and jump straight into something too difficult?
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