Why can't you multiply vectors?

some notes/clarifications! • "I don't like the title" I thought it was fun, and it's a good conversation starter and defines the whole trajectory of this talk! • at 9:55, when I multiply natural numbers - snailcats are not "units" they are creatures equal to 1 so im right and you're all wrong get out of here with your obsession with units!! "but freya it should be 6 snailcats squared" you're square >:( • at 11:42, there's some nuance in how we define closure for division, since you usually can't divide by 0, so technically there's exceptions here, but the general idea stands if we ignore 0 :) • at 25:14, the divine truth bestowed upon us by salad, of v² = |v|², only applies to vectors - it does not apply to bivectors or rotors, or anything else! In my approach, this is a fundamental axiom, which is one way you can define a clifford algebra, in case you want to read up on this some more! It is our starting point, the initial assumption, similar to how i² = -1 is the initial axiom/definition for the imaginary unit. There are other ways of defining clifford algebras, as usual in math, definitions can work from many directions, and the math works out the same way regardless, I just happened to pick this approach for this talk, because I think there's particular beauty in how it so cleanly produces many of the products and constructs we're familiar with! • at 28:27, the reason I'm expanding (x+y)² to xx+xy+yx+yy instead of xx+2xy+yy, is because at this point we don't know whether multiplying vectors is a commutative operation, so we can't say for sure if we can simply swap x and y here. Real numbers are commutative, but in this case we have to be careful, because these aren't real numbers, they're symbols representing our basis vectors! And as it turns out in the end, vector multiplication is in fact non-commutative, as is the whole VGA multivector multiplication • at 28:57, the rule of swapping xy = -yx only applies to orthonormal vectors, like our basis vectors, which are orthogonal, and of unit length, hence the name orthonormal. This rule does not apply to arbitrary vectors in the general case • at 43:40 when I apply the quaternion as a rotation to the cube, it assumes the two vectors a and b are normalized, which results in a unit quaternion (a quaternion with a magnitude of 1), which is what we often use for rotations in games. However, for two general vectors, the quaternion/rotor result of multiplying them together is not a unit quaternion, and is thus not a valid rotation representation

Freya is a god-level instructor. If you can make geometric algebra make sense on stage you deserve an award.

You are the 3blue1brown of computer graphics, I wish you made more videos.

I am an engineering graduate student. Didn't expect much clicking into this video but after watching a few minutes, I found this is gold. This video clears up my concepts in relationship between complex numbers and quaternions (which I see often but don't understand) , where they could come from, and introduced an interesting concept of bivectors. All from one fundamental axiom! The math is beautiful and you elegantly presented it. Thanks a lot!

Was listening to the talk, and at 10:00 I was like "Two snailcats multiplied by three snailcats is obvioously six snailcats squared" ;D

Several months ago, I watched your video on Splines. It was fantastic and has been a great help to both me and my colleagues. Just yesterday, I was searching for some materials on Geometric Algebra. I noticed this video was mentioned in your Discord, but I skipped over it initially since I was specifically looking for content related to GA. However, when I stumbled upon it again in the GA Discord, I figured I should give it a watch! From the get-go, I was really impressed with your presentation style. You have a knack for presenting information! Instead of bombarding us with equations and jargon, you guide us step-by-step, posing questions and then answering them. I genuinely appreciate it! I hope you weren't constrained by time when putting this together. It's just brilliant how the presenter I admire so much is covering topics I'm deeply interested in. And the timing couldn't be better for what I currently need. It's hard to put my excitement into words!

Man, you manage to capture centuries of math progress under an hour. You did color code, box up, and animate the abstracts to concrete representations. It was a very efficient presentation, your work keeps getting better!

I literally got goosebumps watching this. You just explained something to me that I thought would remain a mystery to my grave. I can't express my gratitude.

I would like to validate your feelings by saying that I really enjoyed this talk and that it helped my understanding of quaternions!

For most of the latter half of the talk, I was trying to square the circle with how v^2=||v||^2 doesn't work out for complex numbers, and how that felt wrong with how alike complex numbers and 2d vectors are otherwise. Then at the end, the reveal of complex numbers being rotors instead came and everything just clicked into place. Amazing talk, 10/10

I really, really wanted to understand what quaternions meant but never could actually. Then your explanation came and it all felt like it made sense from the start, especially since quaternions aren't the focus of this talk. Color me impressed (as I always am with your content).

You just single-handedly explained quaternions and why 3D rotation happens in 4D better than any instructor I've ever had. Hat's off to you.

I knew about Geometric Algebra before watching this but even so it was really interesting. The pacing was really nice and the explanations were quite easy to understand at the same time.

Thanks for posting this! Love your math videos so it’s nice to get one of your talks aswell

Now this was just incredible! :D To so cleanly present clifford algebra in a way that is accessible from so many different backgrounds, and make it not only coherent but downright tight, neatly tying together all the different concepts you present along the way, is quite a feat ^^ I feel like my understanding not only of bivectors, but of geometric algebra in general, is so much more tangible now than before the talk. Hats off to you, Freya!

Despite already knowing the concepts you’re talking about, it was really informative to see them arise from one another in such a natural manner! Really well done!

Always amazed by your content. Keep up the hard work for all of us that don't like math!

I just watched this whole video before sleeping. It's surprisingly relaxing, loved the talk

As a physicist, I'm used to dealing w/ quantities (number * unit) so two snail-cats times three snail-cats would be six snail-cats-squared... 🙂

So awesome… my gut response was “yea, you can using the geometric product” and then you deliver. Stuff like this will help usher geometric algebra into the mainstream. Well delivered!