Geometric Algebra in 3D - The Vector-Bivector Product (Part 1)

[@acapellascience]

Hey! Just stopping by to thank you for making this introductory series!

[@Anriuko]

Usually when people get deeply immersed in an activity or a field of knowledge, the time passed and various perspective shifts in between their beginner self and master self effectively distort their view into where they came from, and they tend to explain this new reality of theirs in terms of itself to others who didn't yet take the journey to get there, when in fact they'd need to apply a series of inverse transformations and a "change of basis" to get back to that original space and give directions leading toward the new space in terms of that. It's very common for teachers to get tripped up by this illusion of transparency between two people viewing something from different angles, through different lenses, with different assumptions. It's so common that when someone breaks this pattern I feel obligated to be explicit about it and give feedback. What a treasure trove your channel is. You clearly have an aptitude for teaching, which I firmly believe to be something that can't be learned in a school (only cultivated). Great content, intelligently structured & explained. Not a personality showcase, not entertainment competing for my attention by being loud and obnoxious. Just GOOD STUFF. Don't you ever change.

[@harryhirsch2024]

I am quite convinced that without you I would have never been able to start out. If you lecture somewhere else, please let me know.

[@bonbonpony]

07:32 Another way to see it is as a cyclic permutation: as long as the order of these units stays the same (increasing), the sign stays the same too. 22:22 And from this reason, originally dot product was denoted with "v" to indicate this "grade lowering" nature. That's also the reason why wedge product is denoted with "∧", to indicate its "grade rising" nature.

[@bryphi77]

I want to take the time to comment and thank you for all these great vids! As someone who was always good at math, but never had any formal training past high school, these vids have been so very useful to me in my personal quest for knowledge. Thanks...

[@Math_oma]

+bryphi77 You're welcome.

[@tombouie]

Thks & hopefully you're still up-to-speed on Geo Alg because I have a simple question on-it: Let the vector product AB = a sum of ODD grade components. ?Does AdotB = -BdotA & AwedgeB = BwedgeA ? Ex: If grade(A) =1 & grade(B) =2 ; then grades(AB) =1&3 and AdotB = -BdotA & AwedgeB = BwedgeA Let the vector product AB = a sum of EVEN grade components. ?Does AdotB = BdotA & AwedgeB = -BwedgeA ? Ex: If grade(A) =1 & grade(B) =1 ; then grades(AB) =0&2 and AdotB = BdotA & AwedgeB = -BwedgeA I'm trying to figure-out Geo Alg for an arbitrary Nth dimension space and there's ain't much academia help available on Geo Alg out-there I'm trying to get it down 1st before I start-on Einstein's relativity & quantum mechanics.

[@angeldude101]

Others have already commented about it, but I'm just going to add my own perspective on the dot and wedge products. It seems that no matter what grade the arguments, the geometric product always has a commutative part and an anticommutative part. There also seems to allows be only 2 different grades within the output, one of which is the difference of the grades of the two arguments, while the other is the sum (usually). I can definitely see value in defining the dot product and wedge product with either of these properties. One minor issue I have with defining the wedge product as grade-increasing is that it doesn't work if the expected grade is outside the expected range. While we haven't talked about them yet, a bivector times a bivector would have no wedge product as it's defined here since it would be trying to take a grade 4 component in a 3D space. A bivector times a bivector does however still have a commutative component (grade-1) and an anticommutative component (grade-2), so defining the wedge product based on commutativity would be the only way to preserve the behavior of ab = a*b + a^b for a bivector times a bivector. Things get even worse as you get into 4D as you get more combinations that would result in a grade beyond what the grade-increasing product can access. (and bivector multiplication giving _3_ component grades, two of which are commutative, and 1 of which is both anticommutative and neither grade reducing nor increasing.)

[@Fine_Mouche]

2 vectors can be a bivector if there are not colinear, no ? And 3 vectors can be a trivector if there are not colinear to each others, no ? And what is the difference between a bivector Vs a plane and a trivector Vs a space ?

[@davidhand9721]

I don't like the idea of this dual definition of dot and wedge. Some kind of convention needs to be used to keep the grade of your symbols straight

[@JustNow42]

Does this only work for simple Bivectors or for all kinds?

[@HearTruth]

You are a Gift.. thank you Matheomer

[@BariumBlue]

The part where you declare 1/2(aB - Ba) as doing the dot product between grade1, grade2 objects felt suuuuuuper weird to me. Hmm... If we define the wedge product to be the operation that raises the grade... It seems that the operation will be "plus" when either object is of an even grade, and minus otherwise: grade1 ∧ grade1 = ½ ( e1 e2 - e2 e1 ) grade1 ∧ grade2 = ½ ( e1 (e2 e3) + (e2 e3) e1 ) grade2 ∧ grade1 = ½ ( (e1 e2) e3 + e3 (e1 e2) ) grade3 ∧ grade1 = ½ ( (e1 e2 e3) e4 - (e2 e3 e4) e1 ) grade2 ∧ grade2 = ½ ( (e1 e2) (e3 e4) + (e3 e4) (e1 e2) ) ... This FEELS odd, but Geometric Algebra seems to be full of these kinds of patterns when you extrapolate, such as what you get when you square a unit one/bi/tri/4/5/... vector: 1,-1,-1,1,1,-1,-1,1,1...

[@NoNameEntered]

Are you going to do a video on the bivector * bivector product?

[@JimSmithInChiapas]

I have some materials on GA that may also be of interest. For example, *"Solution Strategies for Geometric (Clifford) Algebra Problems"*: v=RMcqNYmahTU . I'll also be linking to these newest videos of yours on my LinkedIn group, "Pre-University Geometric Algebra". The material that you've covered here is something that I was about to start discussing there myself.

[@arminth4117]

I really learn a lot from your GA videos so thanks very much for them! I just want to note that, at least according to Wikipedia, your definitions for a dot b and a wedge b are swapped at the 22:00 mark.

[@qsykip]

Are you sure? I just checked, and it seems like his definitions are correct.

[@arminth4117]

@@qsykip sorry not the definitions, the line under, the swapping relations, no? dot product is symmetric, a.b = b.a; and wedge product is anti-symetric a^b = -b^a

[@arminth4117]

you know what I'm mixing myself up don't mind me

[@joem8251]

This is exactly what I needed!!!

[@Abominas]

nice video but way too quiet

[@BongoFerno]

21:30 This has to be wrong. The dot product is commutative, and the wedge product is anticommutative. The signs have to be wrong.

[@Math_oma]

For vectors, yes. Remember we're talking about a product between vectors and bivectors.

[@sdmartens22]

Very impressive video; nice touch with the problem part way through! For a video of similar brilliance i was very inspired by Sandpiles-Numberphile video.

[@Math_oma]

+Shannon Martens What problem was that?

[@sdmartens22]

14:13