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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?

#121

Post by mebassett »

Garlef wrote: post #8

While the article is written very nicely, It seems that this is written out of a perspective of some missing knowledge. The basic object that the author seems to be interested in is that of an "algebra over a field" ( https://en.wikipedia.org/wiki/Algebra_over_a_field ). Specifically: Invertability of all elements with respect to the multiplication leads to the notion of division algebra and these have been studied f…

I think the author is interested in clifford algebras, or more specifically geometric algebras, rather than division algebras.

division algebras tend to be quite boring (if they are finite then they are just a finite field; if they are finite dimensional over an algebraically closed field then they are just the field itself. I guess the quaternions are an interested example in the non-algebraically closed case. but I think if you are over something other than R you're really just talking about a field extension)

clifford algebras are a sort of generalization of the exterior algebra one would have encountered in differential geometry and other spaces.

in fact it could be considered a "quantization" of the exterior algebra. as in "quantum groups". which is an entirely different part of maths. but that's not what this article is about.

I think using the language of geometric algebras / clifford algebras in physics as this article does versus the more traditional language is just a matter of taste.

Re: What is the inverse of a vector?

#122

Post by monktastic1 »

thaumasiotes wrote: post #6
Grustaf wrote: post #4

Ah, another geometric algebra evangelist? I can't figure out if GA actually adds anything substantial, or if it merely lets us write some equations in a more succinct fashion. But it certainly looks cool. As to vectors, obviously they have inverses, additive inverses. Since vectors don't have multiplication, there is no multiplicative inverse, but if you define new operations on them, well then that operation can hav…

> As vectors, obviously they have inverses, additive inverses. Since vectors don't have multiplication, there is no multiplicative inverse A vector is pretty much by definition also a matrix, and there is a standard way to multiply matrices. You can define several inverses of a vector that way, though you can't define a unique inverse. The standard inner product is of course also an exceptionally typical way to multi…

> The standard inner product is of course also an exceptionally typical way to multiply vectors, but the concept of an inverse there doesn't make much sense.

Not all vector spaces are equipped with an inner product. The point is that you can start with some simple axioms and build these more complicated things (inner product spaces, algebras over a field, geometric algebras, etc.).

Re: What is the inverse of a vector?

#123

Post by steve76 »

> What is the Inverse of a Vector?

Manifolds, right? A vector takes a value, adds dimensions, and expands the value by spatial definition. A manifold takes a value, adds dimensions, and constrains the value by surface projection.

Vectors are values. Even though they are made up of a bunch of numbers, they are a value like any other input or output. If you have a vector function, continuity works with vector sums just like continuity between input and output. Maxwell uses this, from Gibbs, to describe electromagnetic fields so he could have an equation for empty space and get the speed of light. Sometimes it's more difficult to find a reason for discontinuity, and are forced to assume continuity.

Lie algebra is when you want to describe something with symmetries. Whatever you are describing, such as the insides of an atom, can't be described with exact values, so you use "what things are symmetric" and from there can get a differential equation.

Re: What is the inverse of a vector?

#124

Post by [deleted] »

kevinwang wrote: post #99
tgb wrote: post #58

Earlier quoted context omitted.

Yes, only three. As defined, two bivectors are equal if their areas are equal and if their oriented planes are equal. Therefore two more degrees of freedom are absorbed by taking rotations of the two vectors in the plane.Along with the rescaling the author noted, we're down to three from six.

Interesting, I was confused about the same thing. So the author is not correct when they say a bivector has 5 degrees of freedom?

[deleted]

Re: What is the inverse of a vector?

#125

Post by orangeyjuicey »

dan-robertson wrote: post #75
Garlef wrote: post #8

While the article is written very nicely, It seems that this is written out of a perspective of some missing knowledge. The basic object that the author seems to be interested in is that of an "algebra over a field" ( https://en.wikipedia.org/wiki/Algebra_over_a_field ). Specifically: Invertability of all elements with respect to the multiplication leads to the notion of division algebra and these have been studied f…

The object the author is actually interested in is known as a geometric algebra. One often sees it discussed as an alternative theory for computer graphics or physics as it works well for expressing things like rotations. See here: https://en.m.wikipedia.org/wiki/Geometric_algebra I think it is probably not so helpful to merely think of it like a division algebra, and it is better to stay focused on the geometry. Cur…

hey, by the way, to prove you right: please check the URL of the blog post

Re: What is the inverse of a vector?

#126

Post by mferraro89 »

jefftk wrote: post #49

> The similarities are so striking that we might think of them as "pseudovpseudovectors". But I won't write them this way because I think that obscures their true nature. Written this way it looks like a bivector only encapsulates three degrees of freedom! > Instead, I will use: ... Because it forces us to remember what those coefficients are attached to. Knowing that a bivector contains five degrees of freedom, can…

Author here: I was wrong about the 5 degrees of freedom thing. A bivector has 3 degrees of freedom. I'll correct the text tonight.

Re: What is the inverse of a vector?

#127

Post by mferraro89 »

kevinwang wrote: post #99
tgb wrote: post #58

Earlier quoted context omitted.

Yes, only three. As defined, two bivectors are equal if their areas are equal and if their oriented planes are equal. Therefore two more degrees of freedom are absorbed by taking rotations of the two vectors in the plane.Along with the rescaling the author noted, we're down to three from six.

Interesting, I was confused about the same thing. So the author is not correct when they say a bivector has 5 degrees of freedom?

Yes, a bivector has 3 degrees of freedom. I was mistaken when I wrote 5. I'll fix the text tonight.

Re: What is the inverse of a vector?

#128

Post by mferraro89 »

codeflo wrote: post #50

The writing is cute and the animations are nice, but none of it makes any sense. I stopped reading at > It is important to remember that bivectors have a certain redundancy built into them in the sense that s a ⃗ ∧ b ⃗ = a ⃗ ∧ s b ⃗ s a ∧ b = a ∧s b . We can write them using 6 numbers or 3 numbers, but they actually convey 5 degrees of freedom. Three (real) numbers have three degrees of freedom, by definition. (And n…

author here. I was mistaken about the 5 degrees of freedom bit. Bivectors have three. I'll fix the text tonight. I'm sorry you wasted ten minutes on my nonsense.

Re: What is the inverse of a vector?

#129

Post by mferraro89 »

tgb wrote: post #53
codeflo wrote: post #50

The writing is cute and the animations are nice, but none of it makes any sense. I stopped reading at > It is important to remember that bivectors have a certain redundancy built into them in the sense that s a ⃗ ∧ b ⃗ = a ⃗ ∧ s b ⃗ s a ∧ b = a ∧s b . We can write them using 6 numbers or 3 numbers, but they actually convey 5 degrees of freedom. Three (real) numbers have three degrees of freedom, by definition. (And n…

You're right about that being wrong, and the author makes the same mistake consistently, but otherwise it looks correct. Some steps have details elided where it maybe should have been noted that things were being skipped, but with correct results. I think it's wonderfully written and a great exposition.

hey, if you have time to detail those mistakes I'd be happy to fix them in the text. Can you email me at mattferraro.dev@gmail.com

Re: What is the inverse of a vector?

#130

Post by amatic »

I would say the title should be "reciprocal" of a vector. Right? The inverse of a function (f^-1) has an unfortunate notation equality with the reciprocal (x^-1), or multiplicatory inverse. Or am I wrong?
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