One proof I noticed was incomplete was the one about Multiplicative Inverses. While most vectors can be normalized, the zero vector cannot. The reason behind it is because a unit vector must have a direction. In fact, it must have the direction of the original vector. However, the zero vector has no direction, so it's impossible to normalize it, as any direction can be chose. Another reason the zero vector can't be normalized is because it has a lenght of zero, so in order to normalize it, we would need to divide by zero.
@IcaroCamposdeAPinto
2 жыл бұрын
@@markv785 I never heard of a vector with infinitesimal lenght, but I don't think it would be useful for what I'm talking about. I never said anything about an infinitesimal vector. Only about the zero vector, which has exactly zero lenght and, therefore, no direction. Also, remember that we need to do a division by zero to normalize the zero vector, which is impossible... Or at least impossible without some bizarre combination of geometric algebra with wheel theory.
@milanstevic8424
2 жыл бұрын
@@IcaroCamposdeAPinto it is possible, but the zero vector's direction is undefined, therefore its normalization is undefined as well. yes, when you want to work with it algebraically this translates to division by zero, leading again to undefined result. in my opinion, there is nothing incomplete about this.
@PretTy_Fish
2 жыл бұрын
I believe that's why in the definition of a field, the multiplicative inverse definition requires a multiplicative inverse for each element *except 0.* I think either he deliberately took it out from the video or forgot to put it in the video, but either way, the only way to complete the proof here is to require that we don't need this property to hold for the zero vector.
@thomashoffmann8857
2 жыл бұрын
@@IcaroCamposdeAPinto there is vector analysis. It uses infinitesimal vectors for derivations. Similar to normal analysis.
@arsnova69
2 жыл бұрын
Really nice effort there... But after all the hype about GA, I can't settle with mere vectors... Can't wait to see the upcoming chapters!
@franciscofernandez8183
2 жыл бұрын
What a Well done video! I have really enjoyed watching your approach to introducing vectors. As someone who has teached linear algebra, I have spend a lot of time thinking how to simultaneously teach how to do arithmetic operations with vectors (using components) and the geometrical intuition of those operations. This "all geometry, no components" way of teaching basic properties has been eye opening to say the least. Thanks for sharing all this beautiful work with all of us!
@Bolpat
2 жыл бұрын
I've been in a similar spot, teaching linear algebra. I agree and would go even further saying that using the tuples of real numbers is a misleading example. It's actively harmful because it limits what you think about as a vector. It will catch you at least in the infinite-dimensional case.
@tombouie
2 жыл бұрын
As per your request, a complication; The 0vector has a magnitude of zero & an orientation of { [ 0vector/[|0vector| =0] } which is undefined & thus the 0vector has no orientation. However [0 times the (A≠0)vector] has no mag but does have an orientation. Or something like that.
@tiagorodrigues3730
2 жыл бұрын
Thank you for these videos! I must admit to having some difficulty going about those exercises, since I've been drilled too extensively to treat these properties as axiomata and just postulate them. How do I go about then "proving" something I postulated? Still, even these informal proofs are interesting for the interplay they show between the abstract algebraic system and the geometry of line segments it is supposed to stand for. Cheers!
@predatoryanimal6397
2 жыл бұрын
try book of proof by richard hammack, it is a textbook on standard methods for proofs in math available as free pdf on the internet
@biblebot3947
2 жыл бұрын
You aren’t proving the algebraic facts (those are axioms), you’re showing that arrows are an accurate representative of those axioms
@tanvirfarhan5585
2 жыл бұрын
seriously man you are just doing a great job
@nice3294
2 жыл бұрын
These vids are great
@phoenix4037
2 жыл бұрын
I love your swift introduction video! I'm really looking forward to the rest of this series!
@UliTroyo
2 жыл бұрын
I love this series!
@furballphantomgaming6791
2 жыл бұрын
This is really cool thank you
@user-oj5kg2zl9z
2 жыл бұрын
Not sure how I got here getting lesson from Dango about vectors, but this will absolutely be usefull now as we ať school just started it. Thank you!
@guslackner9270
Жыл бұрын
Another incomplete proof was the proof of Associativity of Multiplication at 11:20. You said that it is clear that the orientation does not change. This is not true for a negative or the zero scalar, the first is the same as a horizontal mirroring and the second is the same as removing direction information entirely. Nevertheless, this property is still true because the zero vector scaled by anything is still the zero vector, and this is the same as its associative: scaling by anything and then reducing to the zero vector. For negative scalars, the negative value will mirror the direction without changing magnitude, so it can mirror the vector before or after that vector is scaled by some other scalar with the same result. Thus, in both of these strange cases, associativity of multiplication for scalars and vectors is preserved.
@andreatriscari7447
2 жыл бұрын
Thank you it is a precious work this that you are doing!
@jbf81tb
2 жыл бұрын
I really enjoy the series and I'm excited to see more, but this section is a little loose. I think in this section you're doing more "defining" than you are "proving". Like you're essentially defining what properties the zero vector has, by asserting that vector algebra should behave similar to regular algebra, or presenting a hole and suggesting what should fill it. Similarly with commutativity of multiplication, you're just saying that by definition, multiplication of a scalar can happen on the left or right. I don't think having definitions is bad. Every system needs its postulates, but I only chafe at treating them like they are being proven. It's a demonstration of self-consistency.
@sudgylacmoe
2 жыл бұрын
I said in the introduction to this series that my proofs won't be good (until chapter 7). You could prove these properties more rigorously, but I don't think I should be super rigorous here.
@jbf81tb
2 жыл бұрын
@@sudgylacmoe I just think there's a benefit to explicitly segregating your theorems and axioms. I don't know the actual theorems and axioms of geometric algebra, so maybe that's already what you've done, but that's not what it feels like as a learner. It feels like you're trying to prove what zero is, which isn't necessary. Maybe you feel that the zero vector is sufficiently different from zero that it needs to be proven, but it can only be different in a pedantic way that you're trying to avoid.
@noya3004
2 жыл бұрын
I think that's because we haven't learned that learning the structure of a "geometric algebra" is a different thing. Yes, he is actually defining it, but he contextualized it with the traditional object called vectors, not actually proving it. With these axioms the structure of a GA will arise as we define the unintuitive "geometric product".
@milanstevic8424
2 жыл бұрын
@@markv785 do you know linear algebra?
@milanstevic8424
2 жыл бұрын
@@markv785 I asked that specifically because there is nothing different in GA with regards to moving vectors compared to LA, as far as I can tell. I am in computer graphics, and well, basically you don't move vectors. And you move them all the time. Well, I'm not a math person and the answer is really complicated. Perhaps to simplify a bit, vectors are like velocities in physics. You can conceptually move them all over the place, and you don't get different velocities. Obviously for technical reasons, strictly speaking you don't want to move them, because they're unmovable by definition, but for conceptual reasons you might want to think of your velocities as belonging to some billiard balls, so you might move them accordingly, draw them as little arrows sticking out of the balls. I don't know if that helps. In reality, however, vectors are defined with regards to their origin. It needs not be a point in space, the origin can be defined by basis. To return to computer graphics, when you work with vectors, you know very well that all vectors start from some origin, and yet you treat them like they don't. For example (I'll denote vectors with ->), if you have some abstract plane you want to define, there are several ways to do this, but one of them would be to define some point A on the plane, and a surface normal n->. This is obviously with regards to some point of origin O. To define A you need a vector OA-> and n-> is just a directional (unit) vector. To move this plane by some delta vector d->, all you need is to find another point A', which you can find by moving d-> from A, but observe that A is a point, not a vector. So you add OA-> and d-> and find point A', which effectively moves the plane by d->. In other words OA'-> = OA-> + d-> Observe that d-> is also an arrow that starts from the origin, but it could've been conceptually drawn at A just as well. Similarly you can draw n-> from the origin, but conceptually it makes more sense to draw it at A, because it shows more clearly where the surface of the plane is. Here's an image to make this more palatable. mathworld.wolfram.com/images/eps-gif/Plane_1001.gif By now you probably worry how to apply a transformation then, when you can't move anything. And yes, normally 3x3 matrices do not actually allow translation at all, just scaling and rotating. But the other important thing is, well, you move points, not vectors. Vectors are always defined in relation to a basis or to another vector. And this is basically a problem with linear algebra specifically, where you constantly juggle between separate vector spaces, and transforming between them, or the results quickly turn into garbage. Typically these transformations are done with 4x4 matrices because they can handle translation. You take a standard 3x3 and slap onto this a 4th column with translation components, and a row of zeroes, ending with an 1. And this is how you hack a common TRS in modern GPUs. 4x4s are collectively used for 3D affine transformations which contain translations. GA, as far as I can tell, and again I'm not a math person, should be more robust than LA when it comes to juggling between vector spaces, at least I'm hoping that's the case.
@gbpferrao
2 жыл бұрын
dude what you doing is amazing, i cant imagine the amout of work, but I can assure you what you doing is very important for the future of math and math students
@keypo790
Жыл бұрын
THANK YOU AxisAngles&RichardPenner😭
@JosephVFX
2 жыл бұрын
It’s unfortunate how you called the equation 𝒗/|𝒗| = 𝒗̂ the “vector version of b/b = 1”, because the equation 𝒗/𝒗 = 1 is much more deserving of that name. I suppose you can’t show 1/𝒗 = 𝒗/𝒗² and hence 𝒗/𝒗 = 1 yet, because you haven’t introduced the geometric product?
@namanmishra703
2 жыл бұрын
As he said in the previous video (when talking about the scalar 0 being equal/unequal to the vector 0), he is restricting himself to linear algebra for now. v/v is undefined in linear algebra.
@sudgylacmoe
2 жыл бұрын
These are all facts about scalar multiplication, not the geometric product. Several of the other properties in this video are different with the geometric product as well.
@keypo790
Жыл бұрын
17:16 from what I recall, dividing vectors has a process of finding its conjugates, when finding a conjugate for the denominator it would lead the complex number to a real number, and the numerator are just going to be foil-method. see vector Z = x + yi, vector W = a + bi Z/W = (x+yi) / (a+bi) multiply the numerator and denominator by (a-bi)... (x+yi)(a-bi) / (a^2 - b^2) then you'll have a new vector your scalar (1/a^2 - b^2) * N
@DMWatchesYoutube
2 жыл бұрын
Having fun so far, can't wait for 1.5
@gabitheancient7664
2 жыл бұрын
you could prove most of these properties by treating scalar multiplication as multiplying each component and vector addition as adding each component, but your way is way cooler
@sudgylacmoe
2 жыл бұрын
What's a component? ;) In all seriousness, it is very intentional that I'm holding off on mentioning components for as long as possible.
@franciscofernandez8183
2 жыл бұрын
@@sudgylacmoe are you sticking to a "direction and length" definition of vectors, so defining bivectors becomes a mostly seamsless generalization? If so I take my hat off to you, good sir. If you are doing it for a different reason, I will probably still take my hat off when I figure it out.
@engelsteinberg593
2 жыл бұрын
And stupider.
@franciscodanieldiazgonzale2096
2 жыл бұрын
He stated the method he will be using as a strategy to teach, like it or not. Doing abstract calculations from the start to solve the problem is only one approach. There is a strategy for a reason and he already explained it.
@keypo790
Жыл бұрын
13:17 When scaling a vector is the same thing when scaling a triangle, basically scaling a resultant vector is just saying to scale the components of the vector( which means the other two the v&w-vectors), and scaling the components of those two vectors before arriving with your resultant vectors doesn't change a thing. Therefore its True.
@DJ-yj1vg
2 жыл бұрын
Very informative
@AdobadoFantastico
2 жыл бұрын
Yaaaayyyy! I look forward to these.
@pablogh1204
2 жыл бұрын
19:38 I don't know if this is the complication you are referring to, but it would be that you only considered the sum of vectors, that it was necessary to consider antiparallel vectors and/or consider negative scalars? Or apply the commutative,... property/ies in the following demonstrations?
@scottdavies161
2 жыл бұрын
I wanted to tell you, I appreciate your style and knowledge. I can tell you work on your discipline for fun. Keep it up!
@spyro1159
2 жыл бұрын
I can’t wait for the next video
@SAJAN_ECE
2 жыл бұрын
This is a great content! Keep the good work! Expecting more!
@lourencoentrudo
2 жыл бұрын
I appreciate greatly a more geometrical and visual way of thinking about it, but there is something about this approach that bugs me. Not all vectors can be represented as arrows. Should these arguments still apply to them? And if so, why?
@sudgylacmoe
2 жыл бұрын
In this sense, the only vector that can't be represented as an arrow is the zero vector. You can verify all of these things for the zero vector if you want. If you are talking in the abstract sense, vectors are defined as objects that follow these properties, so it goes the other way around from what I presented in this video. In those other cases, you have to verify these properties yourself.
@keypo790
Жыл бұрын
14:37 can be thought of distributing (b+c) to a complex number Z = (x + yi) which is (b+c)Z = (b+c)x + (b+c)yi or by (b+c)Z = bZ + cZ distributing Z vector to each of scalars. = bx+byi + cx + cyi = bx + cx + byi + cyi = (b+c)x + (b+c)yi which is the same thing, as distributing (b+c)scalars to each components of our vector Z. therefore (b + c)Z = bZ + cZ is true
@motazfawzi2504
2 жыл бұрын
21:09 Ah Vsauce
@soilsurvivor
2 жыл бұрын
So far, a great series! Thank you! Very tiny grammatical point: you can write "if then ", or you can write "if , ", but using both the "then" and a comma" isn't strictly correct grammar.
@namanmishra703
2 жыл бұрын
Suggestion: Tone down the colour of the vectors and symbols a bit. The blue and red especially are too dark. The yellow and orange are pleasant on the eyes.
@aliassaf8782
2 жыл бұрын
Good work. Please keep going.
@ShadowZZZ
2 жыл бұрын
That's just linear algebra, and a visualisation of the field axioms and vectorspace axioms, and the structures that follow. But you didn't provide any formal proofs which logically follow from only using the axioms, but use intuition. I guess that's nice for laymen people but that's not rigorous LinA
@franciscofernandez8183
2 жыл бұрын
The intend seems to be giving "laymen" (that's the "from zero" part) vector and multivector spaces intuitions. This is an animated youtube series, not a textbook. The approach makes complete sense for the format in my opinion. In fact adressing so many propierties was indeed going "deeper" in the formal sense that I expected.
@chadliampearcy
2 жыл бұрын
Think of this series as 'undergraduate level' rather than 'graduate level'. Less rigor more understanding. One can always do more rigor later in one's career if one is interested enough for that. Rigor is not for everybody nor should it be. Rigor belongs to the minority.
@franciscodanieldiazgonzale2096
2 жыл бұрын
If you didn’t watch the two original videos plus the how to teach video, you are completely missing the point. It will get there, after stretching as much as possible the intuition part, to be able to learn new content, not only linear algebra, in a way compatible with deeper understanding but also longer retention of knowledge by putting an easy path to retrieve the lessons in your head. It is an strategy not available in a half semester course. Also Geometric Algebra is still not available as first year undergraduate, and he also want to reach those students. It is useful for different applications and students that are less interested in proofs, so he will introduce them later, as he announced. The difficult part of the course is not the content itself, it is the form of the course, a strict criteria he already explained in a dedicated previous video.
@keypo790
Жыл бұрын
11:21 Proof Scalar A,B and a Vector C that has a components of x + yi (a complex number) (AB)V = (AB)(x + yi) = (AB)x + (AB)yi and when... A(BV) = A(Bx + Byi) = A(Bx) + A(Byi) therefore (AB)V = A(BV) True..👍
@dihovistos6384
2 жыл бұрын
SPOILER ALERT: my thoughts about the challenge I believe it's about geometrical proofs which assume 2D vectors. I'm not sure I necessarily see an easy way to extend it to more dimensions. I'm thinking along the lines of showing that you can actually do all the reasoning in each dimension independently and when you need length, use Pythagorean theorem to combine the results.
@lumipakkanen3510
2 жыл бұрын
Ending on a cliffhanger I see.
@abcdef2069
2 жыл бұрын
you should teach geometric algebra only like from your other videos from the beginning, i am here to learn geo algebra's basic arithmatic to get used to them
@fahamedi
2 жыл бұрын
Nice! Keep going
@benjaminojeda8094
2 жыл бұрын
I need the next part
@sudgylacmoe
2 жыл бұрын
Sorry, it's coming! It's just been much harder than the previous ones for several reasons.
@arsicjovan9171
2 жыл бұрын
I think the complication is that the absolute value of a function with vector inputs is not proven to be the same as the function of the lengths of those vectors as inputs. Written half - mathematically, this is what is left to be proven: |f(Vector1, Vector2)| = f(|Vector1|, |Vector2|) Edit: I found that this equation is only true if all of the input vectors lie on the same straight line. In that case, we can treat them all as numbers on a number-line, so the equation obviously holds.
@MirkoCrafter
2 жыл бұрын
What about division by the Zero-Vector? :)
@kyleowens3426
2 жыл бұрын
Heya, I was really intrigued by your intro video to geometric algebra, and I intend on following this series through to the end because there are things you touched on in that intro video that caught my attention. However, I wanted to provide some feedback, since you've mentioned that you hope this series would demonstrate how animated video can serve as an alternative to textbook teaching we learn in school. What I wanted to say is that the redundancy starts to cost the attention of your audience. I've noticed that there is a lot of analyzing every possible scenario, which feels like saying the same thing 8 different ways. And once it starts to feel like the video is retreading the same ground over and over, it's hard to stay attentive. The last 3 videos could effectively be condensed into one video of about 20-30 minutes. This video feels like it should have been 6m long max. It also starts to feel after a while that these videos undervalue the intelligence of its audience (as in the audience's ability to take a concept and expand their understanding of it on their own by applying it to other things. For example, if a 2D vector can be expressed as a pair of scalars, then it could be assumed that the properties of scalars such as associativity and commutativity would also apply to vectors, or that if a vector can be expressed as a combination of a scalar and a direction, that scalar properties would also apply to the scalar component of the vector. It may be more helpful to point out cases where that assumption may be incorrect, rather than recover the same principles for every operation). Overall, so far this series falls into the same pitfalls that make textbooks hard to read and understand on their own. And I'm not someone of much mathematical education. My highest level course was college algebra.
@PeeterJoot
2 жыл бұрын
Nice use of color!
@keypo790
Жыл бұрын
10:25 that is only true because the vector in this lesson contains only two components and can be represented in complex number, but if we are dealing with vectors that has three components, when multiplying those kinds of vectors they cannot be commutative any more and order does matter, and is useful represented as 4 components instead and is called a quaternions. And more the reason it could only be true is that the foil-method of the vector components. (which makes more kinda sense --just wanted to add this.
@thomasolson7447
4 ай бұрын
I paused the video. Can I unpause it now?
@blinded6502
2 жыл бұрын
Uh. I mean, you could've just explained what a linear transformation is, no? That there exist multiplication and addition, which we apply to sums of values of different types. Each case of multiplication is described with a matrix. And matrix can basically tell you that you should swap each goose you have for 0.3 of imaginary units. And once you've explained how do matrices/linearities work, you can stop bothering with explaining every case of multiplication/addition you come across.
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