This is so well done. Everything is explained from the ground up, introducing terms and notation not as arbitrary decisions but as reasonable choices and showing the thinking process. I'm really looking forward to explanation of stuff I don't already know.
@catalinmihit
3 жыл бұрын
A nice way to think about scalar multiplication by negative numbers is that the direction (the line on which the vector lies) doesn't change, only the orientation. Eg vectors from 0 to 1 and from 0 to -1 have the same direction (the x-axis) but opposite orientations. I know you defined orientation differently in the series but, at least for me, it's a difference that makes things clearer, hope this helps:)
@tiagorodrigues3730
3 жыл бұрын
Hi, I've just passed through to thank you for presenting exercises, particularly "dumb," simple calculation exercises. I've looked at the exercises in Hestenes's _New Foundations for Classical Mechanics_ and the exercises are mostly "prove this theorem about geometric numbers," which is singularly unhelpful outside of a classroom setting. However, I cannot bring myself to call a circumflex a "hat..." 😉
@debblez
3 жыл бұрын
well in that case you’re definitely the first person to call it that
@tiagorodrigues3730
3 жыл бұрын
@@debblez The second, perhaps; Sudgy calls it "circumflex" before calling it "hat"... 😉
@kijomusic666
3 жыл бұрын
Another great video! Can't wait for the more advanced topics though :) I also wanted to add these videos sent me down a spiral of trying to learn as much about geometric algebra as possible, so thank you for that!
@kashu7691
3 жыл бұрын
check out geometric algebra for physicists by Doran. I highly recommend it
@EccentricTuber
3 жыл бұрын
This video totally has me excited for the future episodes now! Not that I wasn't excited before though.
@eccentricstudent7805
3 жыл бұрын
Hello, other me.
@coolmangame4141
3 жыл бұрын
Seeing how he explains these concepts, I can already tell this series is gonna be a game changer for lots of people who are trying to understand these stuff
@Bolpat
3 жыл бұрын
I remember asking my math teacher which orientation the zero vector has. He answered: “In some sense all directions at once, in some sense none at all.” It sounds stupid, but it really nails it.
@sudgylacmoe
3 жыл бұрын
It's funny, because I initially didn't mention the orientation of the zero vector at all, then a Patreon supporter suggested I mention how it has all orientations, so I mentioned how it has no orientation.
@charlesconnors1066
3 жыл бұрын
I'm loving this series. Keep 'em coming!
@nilsh5027
3 жыл бұрын
first edit: I've never been that early on a video before, so I had to lol. But that was a great explanation of the notation! It would've been a cool (albeit rather trivial) exercise to see what happens when a scalar goes through the Pythagorean theorem just so that it would be clear that scaling works even for vectors that aren't on an axis.
@sudgylacmoe
3 жыл бұрын
I had just clicked to publish the video, then needed to get the link to it to post it on other sites. When I went to get the link, I was shocked to find a comment!
@angeldude101
3 жыл бұрын
Holy cow, I just watched the first two videos either yesterday or the day before. Seeing this come out so quickly afterwards was pretty exciting.
@SAJAN_ECE
3 жыл бұрын
Amazing content! Thanks for starting from basics!
@kodirovsshik
3 жыл бұрын
So glad to see another episode on geometric algebra from you!!
@farvardinmainyu1961
3 жыл бұрын
Thank you sudgylacmoe !
@onafets38
3 жыл бұрын
I really appreciate your firsts videos of this course.. but maybe this is the video that I like the least. I have also to recognise that maybe it’s the hardest video to make.. this is because for this chapter so “preliminary” it is hard to take it separated from what will come next or from the foundational topics of mathematics.. it’s hard also maintain in this case an approach based on intuition etc.. (A sort of concrete and practical approach that you have in your previous videos and that I liked).. That said, i m eager to see the next chapters .. keep up with the good work! bye :)
@fahamedi
3 жыл бұрын
Can't wait for the next steps towards geometric algebra
@ReginaldCarey
6 ай бұрын
I have to break in here. The justification for the change in orientation is that the - sign really represents a unit vector with a \pi rotation. In other words in addition to scaling the vector, the minus sign applies a rotation i.e., the vector is multiplied by e^I\pi. I wish the concept of negation was always described in the correct context. In terms of representation I propose that the numeric representation of a number be ae^ib . Were b is often seen as \theta a pain to type. The number Zero COULD be viewed as a=0 as the only requirement. e^ib is the e1 unit vector. Note that negative numbers are not necessary in this notation b is between 0-2\pi. Real numbers have a bimodal value of b ={0,\pi} Is it more verbose? Yea but it immediately puts to rest the concept of a scalar.
@emersonchaves567
3 жыл бұрын
Great series and a great video. Keep it up!
@tethyn
Жыл бұрын
Another good video. Scalar multiplication changes sense but not orientation of vector. Or if you want to just include sense and orientation together and say that scalar multiplication does change its direction. I prefer the first way and be a little more exact with what direction means. Also Pythagorean theorem will always work; however, it changes from the A^2 + B^2 = C^2 into a function (specifically the metric) that shows how vectors are measured in non Euclidean spaces. That being said… nit picking… still gets the main ideas across that is necessary to get where you are going. Field properties are being introduced without stating but I am sure you will get to that when necessary. Good job and take care.
@eustacenjeru7225
Жыл бұрын
Excellent
@ameer.a_r
3 жыл бұрын
oh god what a masterpiece this series is!!
@alamagordoingordo3047
3 жыл бұрын
Great job.
@tonyvanoost
Жыл бұрын
I love your video lectures! In 1.2 you say that the vector zero defines the origin. In later scalar multiplication exercise, you say that 0 v4 is located at location (4,2). Can you explain please.
@sudgylacmoe
Жыл бұрын
Vectors don't define points, they represent points. The way they do that is by using a vector that starts at the origin and goes to the point they represent. The location of a vector has no bearing on what the vector itself actually is, so the zero vector that starts at (4, 2) is the same thing as the zero vector that starts at (0, 0).
@anuman99ful
3 жыл бұрын
These videos are unvaluable
@ShadowZZZ
3 жыл бұрын
I hope this is not gonna be any typical intro to vector calculus or linear algebra, but an intro to geometric algebra
@MH_Yip
3 жыл бұрын
Finally, new episode :D
@Patrick_Bruno
3 ай бұрын
The vectors you are describing belong to real vector spaces (elements of R^n). Does it mean that complex vectors (elements of C^n) are not considered in your construction of geometric algebra?
@sudgylacmoe
3 ай бұрын
In this entire series up to chapter 7, I will only be considering geometric algebras over the real numbers. This is because they are by far the most useful, and because geometric algebra often can be used in place of complex numbers, making them much less necessary. However, if you really want to, you can use any commutative ring as your scalars, which I'll be doing in chapter seven.
@pacificll8762
3 жыл бұрын
Awesome !
@geordonworley5618
3 жыл бұрын
Wait a minute!
@Apollorion
2 жыл бұрын
In the practice I concluded that minus three times v3 was equal to v1, and that 3/2 times v5 was equal to a vector from O to (-3,-3) . But sudgylacmoe's video does neither agree nor disagree with this. So am I right or did I made a mistake there?
@sudgylacmoe
2 жыл бұрын
Yep, that's right.
@mastershooter64
3 жыл бұрын
How do you represent a bivector or a trivector numerically? Like for example you can represent a vector as a list of numbers [1, 2 ,-4] or like this 1i + 2j -4k (but the letters have like hats on top of them) But how do you do that with bivector or trivectors or tetravectors upto k-vectors? and how do you plot bivectors or trivectors or k-vectors on a graph? you can just draw a line from the origin to the (1, 2, -4) to draw a vector on a graph, but how do you do that with a bivector or k-vectors?
@sudgylacmoe
3 жыл бұрын
You use unit k-vectors. For example, in three dimensions, you can have three unit bivectors that represent three orthogonal planes in 3D space, and then you can represent any bivector as a linear combination of those three bivectors. I'll be covering this in chapter two.
@mastershooter64
3 жыл бұрын
@@sudgylacmoe ah that makes sense, but how do you represent them numerically so that a computer can do calculations with them, I'm asking because I wanna write my own geometric algebra library(probably in c++) for a nD physics engine that I'm eventually going to write. You can represent a vector as an array of numbers like [-3, 9, 7] how do you do that with a bivector and k-vectors?
@sudgylacmoe
3 жыл бұрын
It's exactly the same as with vectors, you use a list of numbers. Each number corresponds to the coefficient of one of the basis k-vectors.
@mastershooter64
3 жыл бұрын
@@sudgylacmoe ah right, got it, thank you!
@tombouie
3 жыл бұрын
Thks
@stephaneduhamel7706
3 жыл бұрын
At 4:20, you say that we should use the pythagorean theorem to get the length of the vector, but you didn't clarify why should we do it. From my current knowledge of linear algebra, I know we could use other metrics where we don't use the pythagorean theorem for the length, for example by using the sum (or the maximum) of the absolute values of the sides of the triangle as the length, whithout breaking any rules of algebra.
@sudgylacmoe
3 жыл бұрын
I'm not going to be covering other metrics until chapter five. The question "what is the length of this line segment?" is a purely geometric problem, and the Pythagorean theorem is the solution to that problem. Remember that I haven't even covered abstract linear spaces yet.
@stephaneduhamel7706
3 жыл бұрын
@@sudgylacmoe Yeah that makes sense, I just think it could have deserved a line about the pythagorean theorem being the intuitive way of doing it, but not the only one. I'm really enjoing this series so far though, even if we're still at the part I already know, cant wait for the more "geometric" parts.
@1337w0n
3 жыл бұрын
The red one honestly looks like 2 and 3/4
@andreizonga4611
3 жыл бұрын
People say "X channel is the definition of quality over quantity". Well, I can say that this channel is the definition of quality and quantity! Your videos give such a 3boue1brown vibe, but unlike 3Blue1Brown's, they are published pretty often! Kepp up the great work!
@R.B.
3 жыл бұрын
The length of a number shouldn't be explained as absolute value, it should be quickly explained as the Pythagorean theorem applied to two vectors, one on the x-axis and one on the y-axis with a length of zero. Thus abs is a special case of sqrt(x^2+0^2) which for natural numbers just removes the sign.
@angeldude101
3 жыл бұрын
What y-axis? Real numbers only have 1 axis, so trying to add a component that doesn't exist doesn't make a whole lot of sense. In general, there is a common pattern for finding the "length" of a real number and of a vector, as well as other geometric objects, but that pattern shouldn't require jumping into a higher dimension to find.
@R.B.
3 жыл бұрын
@@angeldude101 it doesn't require the y-axis, but it demonstrates why length is positive. A y-axis of zero dimensions is like a z-axis for a Cartesian plane. You don't need the third axis, but it also demonstrates that a zero magnitude in another plane doesn't impact finding the length. Absolute value is the square root of the sum of the squared magnitudes in each of those axises and it works for any of those dimensions, including 0-dimensional spaces I suppose.
@APaleDot
3 жыл бұрын
I disagree. The absolute value is often defined as √x² which is the Pythagorean formula in one dimension. Explaining length as an absolute value is not only correct, but gives the students something familiar that they can use to anchor themselves.
@R.B.
3 жыл бұрын
@@APaleDot if a student has never heard about Pythagoras theorem, then I'd agree with you. I think more students are taught absolute value that if a number is negative, make it positive. So all I'm trying to suggest is a way to explain why the magnitude should be calculated as sqrt(x^2) rather than just removing the sign. If students already understand how to calculate the hypotenuse of a right triangle, then I think it is worth while to show that this is why the magnitude is not negative. Absolute value is a special case of applying the Pythagoras theorem to a single dimensional vector and makes it easier to understand in higher dimensions, if only as a bridge of knowledge to provide greater understanding.
@de_soot
3 жыл бұрын
thanks man, very cool
@arpanbanerjee5659
3 жыл бұрын
made with manim?
@sudgylacmoe
3 жыл бұрын
Yep, that's what it says at the end.
@porky1118
3 жыл бұрын
I already thought, we can't take Pythagoras for granted and have to find out why it's a²+b²=c² and a³+b³=c³ or something like that.
@DiThi
3 жыл бұрын
I recommend Mathologger video about proofs of pythagoras. And as far as I know, it's always squared because each sum occurs in a 2D plane (defined by the 3 points of the operation), and combining more than two sums is the same as doing the theorem several times in sequence, in any order.
@user-ur2dn7xu3w
3 жыл бұрын
Your videos are graet!!!, when you start to publish a video on geometric algebra and not on liniar algebra. I think most of the people who watch you know liniar algebra, you can do a Survey in your channel. In general it's interesting what the level in math of the people who watch the channel And soory abuot my English
@sudgylacmoe
3 жыл бұрын
The point of the series is to make geometric algebra more accessible for high schoolers, so covering vectors is pretty much required. Also, I've been surprised at how many people I've seen trying to learn geometric algebra without knowing much about vectors.
@deltalima6703
2 жыл бұрын
4 years of uni. Compared to high schoolers I am an expert, compared to alon amit I am a moron.
@federook78
3 жыл бұрын
I think if we're watching this we already know what vector addition means
@Dayanto
3 жыл бұрын
Part of the point of this series is that the conventional method of teaching this kind of stuff is confusing and unintuitive. Therefore it's meant to be accessible also to people who haven't yet learned about vectors, complex numbers, and so on.
Пікірлер: 65