Geometric Algebra in 2D - Some Trigonometry

In this video, we will derive the law of sines and the law of cosines using the concepts of geometric algebra, namely the wedge product and geometric product. We'll also review some other facts about the geometric product and the mathematical objects that we have been discussing so far in the geometric algebra of R^2.
Another video series covering Geometric Algebra (Nick Okamoto): • From Vectors to Multiv...

[@MultivectorAnalysis]

These are great! Thank you so much for doing a series on GA. This is a subject that should be much more well known than it is, and things like these introductory videos are precisely what we (the GA community) need more of. About a month ago I posted a GA tutorial (see my youtube page if you're interested) that stemmed from a graduate seminar talk I gave a few years back. Though my GA vids could serve as a first introduction to GA (probably requiring liberal use of the pause button to allow the rapidly presented new concepts time to settle in), your videos provide a gentler introduction to the subject. I'll be including a link to your GA playlist in my video description. Keep up the good work!

[@Math_oma]

+Nick Okamoto Yeah, it seems like a topic that doesn't get enough attention and connects together a lot of concepts from (seemingly) different areas of math. I noticed that people on KZbin have made videos on the subject before but they seem to be at too high a level of abstraction, whereas it doesn't seem to me one needs to know any more than a few linear algebra concepts to understand the main GA concepts. Of course, there's nothing incorrect about starting abstractly first, but I guess it depends on your target audience, right? I think I've seen your videos before when doing the preparations for making my own videos. I'll also link to your GA playlist, as it's always good to see different approaches to the subject.

[@MultivectorAnalysis]

Agreed. The minimalist axiomatic approach I take in my GA vids demonstrates how natural the geometric product is (as an answer to how we should define the product of vectors), leading inevitably to the multivector system of GA. But it was designed with a graduate level audience in mind. I was considering making a more undergraduate friendly introduction to GA (starting, as you do, with the inner and outer product), but I'm just too busy finishing up my dissertation this year. That's why I was so grateful to find that your very well done videos took care of that. I am planning to make more GA vids in the future, and now I no longer have the dilemma of deciding between advancing from where I left off or going back and presenting a more introductory approach.

[@prabhatpal3374]

Good job with your videos, Nick!

[@connorfrankston5548]

Indeed those parallelograms must have the same area, since the same triangle has exactly half the area of each, but it is nice to see the result produced algebraically by the exterior product. I especially liked the law of cosines deduction!

[@walterufsc]

Absolutely fascinating! Thank you so much for these great videos.

[@osamahasan3288]

Excellent lectures. thank you.

[@cristian-bull]

these videos deserve more views

[@mischa1234]

you are a boss, keep up the good work.

[@ChatterboxBS1]

So nice to see someone finally making khan academy style videos about geometric algebra! I do have a couple of questions. You derive the laws of cosines, sines and Pythagoras here. However in lecture 1 your started with two orthogonal basis vectors. And you mentioned that orthogonality means the dot product must be 0. But the dot product requires coordinates of a vector to have meaning. This means that the dot product, unlike the wedge aka outer product depends on the choice of basis vectors. I always was told that geometric algebra was coordinate free. I think what I am missing in the GA picture is a clear understanding of the concept of a metric, and how the GA in one basis relates to the GA in another basis. Can you shed some light on this? Also, how should we see vectors? As arrays of numbers? Or as geometrical entities existing in some absolute mental universe? If we choose the latter, how to actually determine the coordinates of one vector to another then? I know mathematically this is all solved by just defining axioms and laws, but I always feel some chicken or egg problem exists both in the definition of GA or regular vector algebra. I mean you can't speak of length of a vector without having a basis. But in the abstract axiomatic definition of GA, this is done, the geometric product is defined as the square of a vector being the square of its length, or dot product with itself. I feel it hard to explain my dillemma ;-)

[@miroslavjosipovic5014]

This is a nice question. We do not need coordinates to define the concepts you mentioned. First, the concept of orthogonality is not connected to coordinates and dot product. Note that we can always write ab = (ab + ba)/2 + (ab - ba)/2. We can say that vectors are parallel iff they commute, and orthogonal iff they anti-commute. There is no need for a metric here. Consider now the product a a, where a is a vector. It easy to show that this object commutes with all elements of geometric algebra. If ab = -ba (the vectors a and b are orthogonal), we have aab = -aba = baa. Now we have square(a+b)= aa + bb +ab +ba = aa + bb. This means that we have the Pythagorean theorem for every signature.

[@okuno54]

I know it's an old comment, but "you can't speak of length of a vector without having a basis" stood out to me. If I draw some random arrow on a sheet of paper, I can grab a ruler and line it up with the arrow, and that's how you measure its length. I don't need to have a basis to do that. Heck, I don't even need numbers! Instead of a ruler, I could line a compass up with the endpoints. Then, I can move the compass over to some other arrow to compare their lengths: if the second arrow doesn't intersect with the circle I can draw based on the length of the first, the second arrow must be shorter than the first; the other relational operators can all be defined in terms of 'less than'.

[@majuraselekwa172]

Is there a better way of justifying that e_Ae_B=e_Ae_C=e_Be_C on deriving the Law of sines

[@jamesking2439]

Wow! That's very elegant.

[@hoodeinstein4415]

Man... I just want to know. How long did it take you to study geometric algebra? Why, what motivated you to study this subject? And what are you using it for outside of education? I really hope you get back to me as I am really curious about your personal opinion on these questions.

[@Sidd-rb4ec]

loving the series , just started, can you tell me the software you use for wiriting

[@Math_oma]

SmoothDraw

[@Sidd-rb4ec]

@@Math_oma also can you help me with the intuition for why we get x and y in the linear eqns as ratio of area, why should they be the ratio of areas , also which book or material did you follow to make the lectures

[@DrDoubleplusungood]

In the law of cosines part, if I start instead with C = A + B and go through similar steps, I get c^2 = a^2 + b^2 + 2abcos(θ) (which is wrong, isn't it?). But what mistake am I making?

[@Math_oma]

+DrDoubleplusungood There's no mistake, it's just that θ isn't an interior angle, rather an exterior angle, which you'll notice if you draw out the triangle, making sure to consider the angle between A and B joined at their tails in the dot product step. This means that the interior angle is really pi-θ and this will introduce a minus sign into the cosine function, since it flips sign upon shifting pi radians.

[@DrDoubleplusungood]

I see, I forgot to connect the tails before considering the angle. Thank you!

[@rasp1628]

hi