Why can't you multiply vectors?

Freya Holmer, of The Continuity of Splines [0] fame, recently gave a great conference talk about how quaternions naturally arise from the desire to define the multiplication of two vectors if you just agree to one new axiom : v^2 = ||v||^2, or "the square of a vector is equal to the square of its norm". It was fascinating and entertaining and I highly recommend anyone who finds this stuff interesting but confusing (like I do) to check it out: [VIDEO] [0] [LINK]

Coincidentally, this conference talk[1] released yesterday regarding vector multiplication. In it, it describes how quaternions can be seen as an extension of 3d vector spaces, like how complex numbers can be seen as an extension to 2d vector spaces. More specifically, quaternions are a 4d vector with the basis of {1,yz,xz,xy} where those "bivectors" result from 3d vector multiplication, e.g. (ax+by+cz)(dx+ey+fz) [1]: [VIDEO]

I know people have jumped here to try to explain it to you as a comment, but I highly recommend a recent talk by Freya Holmer. It's a fun little exploration of a seemingly simply question about vectors that motivates the answer you asked. It also culminates with an introduction to Clifford Algebras, which you may or may not want to learn about after the talk. [VIDEO]

True, but he did use quaternions (by 1873), which allow the field properties to be written as a single equation. It's kind of sad that more physicists don't use or teach quaternions, while math and CS have fully adopted them. I really liked Kathy Joseph's historical reviews of vector physics and the people who developed it, which explain some of the reason's for how it's taught. Most texts don't even develop electrodynamics from relativistic electrostatics as a demonstration. [LINK] I think I fell down the rabbit hole from Freya Holmer's "why you can't multiply vectors". [VIDEO] The key being that all of the Hamiltonian fields can be found in a single quaternion equation, which is just what happens when you start multiplying vectors together.

I got sucked down a Geometric Algebra rabbit hole a few months ago, which seems like a remarkably concise and intuitive way to work in a very wide variety of geometries, including 2D, 3D, 4D+, non-euclidean etc. I've wondered if GA would make a good foundation for a physics engine... There are a few small rust libraries that look interesting [1][2], but none with a lot of traction. Has anyone looked into this? [1]: [LINK] [2]: [LINK] If you dare, a good place to jump in might be Freya Holmér's Why can't you multiply vectors? - [VIDEO] - Then [LINK]

One of my favorite math/graphics YouTube creators Freya Holmér did an excellent intro to Geometric Algebra not that long ago [1]. If you have any interest in 3d graphics (especially but no limited to splines/Bezier curves) then be sure to check out all of their videos. I personally have always struggled with linear algebra and I tend to find these Clifford Algebra approaches much more intuitive. 1. [VIDEO]

In another comment, they pointed out [VIDEO] in which Freya did an excellent job reducing the number of axioms to 1: If you define multiplying a vector by itself to be equal to the length of the vector, squared, then everything else falls out of simple polynomial multiplication. It's quite lovely.

If you are still interested, try this: [LINK] The idea is that quaternions are the wrong way to look at the problem and that a better approach would be to use geometric algebra, bivectors and rotors. The formulas are essentially the same as with quaternions, but at least for me, this approach make more intuitive sense. It also work in dimensions other than 3, which matter to the author as he is the author of "4D toys" and hopefully, Miegakure. Here is another talk on the subject, by a different author: [VIDEO] It is not specifically about quaternions, it is about multiplying vectors and what you get from that, and it includes quaternions.

You should jump straight to geometric algebra. It's actually simple there. Quaternions and Complex Numbers are about just some not so special multivectors. Here are 2 45mn videos you'll not regret: [VIDEO] [LINK]

The absolute first video to watch should be the (ironically titled) Why can't you multiply vectors? [VIDEO]

Thanks for mentioning this. I think I encountered Geometric Algebra in a video related to making video games. It's this one (Why you can't multiply vectors): [VIDEO]