Why can't you multiply vectors?

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: ▶ this video It is not specifically about quaternions, it is about multiplying vectors and what you get from that, and it includes quaternions.

in “Quaternion Knowledge Graph Embeddings (2019)” · View on HN

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? - ▶ this video - Then [link]

in “Rapier is a set of 2D and 3D physics engines written in Rust” · View on HN

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]: ▶ this video

in “Exploring Quaternions (2018)” · View on HN

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): ▶ this video

in “Functions Are Vectors (2023)” · View on HN

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". ▶ this 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.

in “Why is Maxwell's theory so hard to understand? (2007) [pdf]” · View on HN