Cross products | Chapter 10, Essence of linear algebra
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- Опубликовано: 30 авг 2016
- This covers the main geometric intuition behind the 2d and 3d cross products.
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*Note, in all the computations here, I list the coordinates of the vectors as columns of a matrix, but many textbooks put them in the rows of a matrix instead. It makes no difference for the result, since the determinant is unchanged after a transpose, but given how I've framed most of this series I think it is more intuitive to go with a column-centric approach.
Full series: 3b1b.co/eola
Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced.
3b1b.co/support
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Vietnamese: @ngvutuan2811
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These videos are just so damn good. Kudos to you Grant, keep up the great work!
+patrickJMT Thanks Patrick! Keep up the good work yourself.
+3Blue1Brown I'm a big fan of you two :)...
patrickJMT I'm a big fan of both of you. As a physics student, Patrick has come to my rescue many times at 3 am during homework binges lol
Two great Math tutors! Thanks a lot to both of you. We wouldn't be able to understand well without you.
Collab!
"The order of your basis vectors is what defines orientation."
Oh man, that was just one quick sentence and I feel like I understand linear algebra so much better just because of that.
I agree :D It's nice because it translates the geometric notion of orientation into other vector spaces (like the space of N x M matrices over some the integers or reals, or the space of functions from R to R). I found this a cool concept too!
English is not my first language and I can't find a translation for this sentence that makes sense to me :(
@@juliaprohaska9295 (German translation:) "Die Reihenfolge deiner Basisvektoren bestimmen die Orientierung."
Why do you use the right-hand rule? Because you're using a right-handed coordinate system (i x j = k).
@@yashas9974 woah
If you ever wrote a text book on this (or any branch of math, I'd wager), it would likely become the standard text for almost all classes on that subject.
a lot in textbooks is also politics. a lot of schools have deals with publishers, teachers have relations with the autors (if you write a textbook you probably know some ppl in the field). there might be waaaay better textbooks than they sold you but they either didnt know them or had an interest to sell you the other one
@@SimonWoodburyForget Ok, so basically we need textbooks with videos _inside_ them.
@@jorgejimenez4325 so its a notebook w/ batteries
I think this series is the textbook.
Sexy black backgrounds would cost too much to print
What I'm now thinking is: Actually making those 3d animations yourself would be a great exercise for programming students or other technical students who have to learn programming along the way.
You sort of project 3d lines onto a 2d-plane, which is your monitor; and that feel of what you doing will coincidently also help you understand linear algebra a little bit.
Ok up for ze challenge.
Yes, he could make a whole class on programing the way he does
@@juandanielcastanierrivas9545 Btw, there actually is something like this on YT...
What I found out last year, is that graphics shaders are programmed with the same matrix multiplications, dot-products, cross-products and vectors that you see in linear algebra.
A dutch guy from the youtube channel "the art of code" tells you step by step how to make 3d effects in shaders using just code.
@@sasjadevries very informative channel, thanks man
I actually only learned about matrices and linear algebra, when I was around 14yo, when I tried learning OpenGL to make games. In a way, it's the best homework exercise possible. It even helped me with chemistry classes, since we were learning linear systems and Hess' Law for chemical reactions, and on maths, we were learning this to solve problems related to logistics, money etc... My professor was awful at teaching matrices, and I was struggling in four different subjects(Algrebra, Geometry, Chemistry and Physics). OpenGL saved my year. I even tried learning General Relativity to create a simulation of the Solar System, repurposing the matrices library of the OpenGL to simulate metric tensors for the calculations. My GPU couldn't handle it at the time though, It was a really old PC after all and I didn't have money to buy one with a good GPU at the time, so the project ended.
I remember going crazy in school because nobody would tell me why cross product is computed as determinant of the two vectors. I can't thank you enough for the insights your videos provide. Thank you 3Blue1Brown. You are amazing.
"That was technically not the cross product"
*angry pi
can you state the reason for your compliment
@@prasannapk6181 at 5:05 the blue Pi in the middle gets angry when 3b1b says that's not the cross product
I'm the chill pi
I was going to like your comment but the number of likes were exactly 314 and the pi reference stopped me from doing that!
But I'm not pi
These videos are getting weirdly addicting
Oh hey there! I'm watching this now to learn vector xD
Ikr
Yuesh
I love how your series can serve both purposes: an introduction for beginners and a reinforcement for students who know about the computations but not the intuitions
the second one is me lol
This is the best intro to Linear Algebra I've ever seen. While I know the maths behind most of it, the visual intuition this course adds is incredibly helpful. Excellent work.
This series has been amazing for me. I took linear 3 years ago as a freshman but, like many people who took it, I really only got the numerical explanation of everything and had no intuition for what was happening and, more importantly, WHY it was happening (which is kind of important as a physicist). These videos are not only a good refresher on linear concepts for me, but brings to light a whole part of linear algebra that I unfortunately never got to enjoy the first time around.
I know you are trying to keep this series as light as possible (the "essence" you may say) and therefore don't plan on going too far out of the basics, but do you think you would do a video on the divergence and curl of functions and give an explanation of how those work? I know far too many people who don't have the best intuition for what's going on with those operations. It isn't exactly a linear concept so I'm more curious if you have future series like this planned that may include those concepts (a calculus series perhaps).
Keep up the great work!
Thanks for the request. You might be interested to know that I've made some (more casual style) videos about multivariable calculus for Khan Academy, which includes div and curl. I will probably do a video on that particular topic on this channel at some point, but I cannot say when. I am planning on an essence of calculus series, but again, I can't make promises about the specific timing.
I had to go check out a sampling of those videos since they must be relatively new. From what I saw, they certainly look like a great learning tool and I would recommend anyone struggling with multivariable calculus to look into them for some support.
As for an essence of calculus (or any other essence of _____), I do find these series very helpful for those who might just need a refresher or a mild supplement (not necessarily a whole class). Don't rush it, however. Your quality is one of the best I've seen in terms of mathematical videos and you shouldn't dilute that for higher quantity. If it gets done eventually, then that is more than good enough for me.
Oh my, an essence of calculus series would be amazing. This series has already been a huge success. I would be saddened if teachers didn't make use if this series, even if they just told their students to go watch as a supplement.
i'm really curious on you've developed these intuitions. amazing and extremely helpful videos!
I sent a link to this series to my Linear Algebra teacher last semester. Sadly, she chose not to share it with the class. Now I'm in CalcIII, and next session my teacher with a thick russian accent will use the word "Parallelepiped".
0:00 intro
0:40 cross-product (2D)
2:15 how to compute?
3:06 the determinant
4:57 the true cross-product (3D)
6:45 formula vs. determinant process
7:50 outtro & note on next video
Over the past year, I've had the pleasure of watching these before my school taught their respective subjects. I watched lockdown math, then after a couple months watched the essence of Calculus (my school taught it to me around 2-3 months after, and it was baffling what a difference intuition can make)
And now, I'm learning Linear Algebra. It's fascinating!
Just want to say I took linear algebra last year, and while I felt I had a decent understanding of the concepts these videos have not just recapitulated everything I've covered, they've really pushed my visualisation and understanding in a way I couldn't reach with just the materials provided. So thanks, I'm glad I found this channel, and can't wait for more content (particularly analysis). Keep up the great work
OMG! Your animations are amazing! I wish every professor explained things so visually and clearly as your videos. You explain here everything comes from, this helps tremendously when learning these "abstract" concepts. Thank you!
A true role model for what teaching online is becoming.
Bravo and thank you!
I almost done translating all of this series to Hebrew CC. Please approve this CC as well.
Yours truly :-)
How do you do that? I mean, either you contact the autor of the video or RUclips?, in order to offer them to upload your subs.
@@germanvazquez7452 I did contact him. I am the one who explained to him in the first place how to make it available to everyone to add CC(When he had less than 1K subs). You need several people from your own language to approve your CC, in order to get approved(As if not approved by the author himself).
Great . @@user-ox2up1yl2g Thanks for making the time to answer me. Greetings from Mexico.
Wow, really? thanks that made Arabic CC possible. Much appreciated it.
@@AhmedMahmoud-tv9vw Most welcome!
I watched these videos to get and understanding of linear algebra last year and wrote off the part about i, j, and k hat. Now I'm in calc I'll and it is there! Truly a brilliant series. You find so much more than what you are looking for in these videos and it's greatly appreciated.
This whole series is brilliant! Thank you so much for putting this excellent material together. It was clearly a lot of work.
This is one of my favorite series! I love the visualization I get from these videos. Great work and keep on making awesome content!
THANK YOU SO MUCH FOR THE BEST EXPLANATIONS .I AM IN HIGH SCHOOL AND NEVER QUITE UNDERSTOOD THE MATRICES AND VECTORS BEFORE AND HENCE I ALMOST HATED THESE TOPICS .BUT NOW IT HAS BEEN THE BEST ,CREATIVE AND FUN TOPIC . I DIDNT EVEN KNEW HOW THE TIME JUST PASSED WHILE LEARNING THIS AND YES IT WAS AMAZING TO FIND MYSELF THIS VERY MUCH FOCUSED . THANK YOU
This is the classiest and most well done youtube channel there is. Congrats!
I just really want to let you know how much I appreciate the production on these videos. It's by far the best editing and video style I've seen in the genre. It's wonderfully focused on the beauty of mathematics and it makes me extremely satisfied to know that there are people similarly interested in it. Excellent channel, I think you have great things to come in the future.
The god of math blesses you, Grant! Finally, I had the answer I was looking for since my engineering studies... an answer I wasn't able to find until this video. Thanks a lot for your contribution to the world of knowledge.
I am writing from Russia. You have good graphics and storytelling. I wish all the lessons in our schools were as interesting as your video!
Wow. This is most comprehensive explanation on cross product I've ever seen. Great channel, mate! It helped so much to wrap my mind around this concepts!
HOLY SHIT. THANK YOU SOO MUCH 3BLUE1BROWN. LOTS OF WEIRD THINGS SUDDENLY MADE SENSE . Like flemings LEFT HAND RULE IS JUST THE RESULT OF THE VECTOR MULTIPLICATION OF CURRENT AND MAGNETIC FIELD RESULT OF THE RIGHT HAND RULE OF CROSS PRODUCT. I WOULD LOVE A TEXTBOOK BY YOU : )
man this channel is so awesome. math is so awesome
What an intuitive approach to math is this!!!!
I just solved hundreds of problems using geometrical approach of this animation....
Thank you so much grant sir
OH MY GOD! I FINALLY FOUND THIS! I used to watch this on KhanAcademy, but it seems they took this playlist off. Thank you for this. Really. They're amazing, and give so much intuition to so many unanswered, and boring statements.
YEEES! I love this series
This video is so clearly through the concept!
Thank you! Second year compsci student here. In our linear algebra course we have only been dealing with calculating and It's really interesting to see how this all looks geometrically.
This reminds me - we need some more Clifford and Geometric Algebra videos in this RUclips circus.
^^^ Exactly!!! You did a great job btw. but Grant if you read this, please help making GA more popular !!!
When i'm linear algebra teacher, could i use these videos to teach? I'm from Brazil, so my idea is talking while your video is playing without sound... My students doesn't know english :p
Samuel Andrade you could translate the video ask someone to translate the video for you.
I think Grant is kind enough to make these vids copyright free. But I think it'd be cheating him if we use these vids for free cuz he puts a lot of effort on them. So he should get money for them. Or you can advise ur students to see his vids with Portuguese captions.
@@randomdude9135 Lol, there's no brazilian, they speak portuguese.
@@user-xj9re7gv5g Ok, my bad😅
There are Portuguese subtitles! :D
Hey, just wanted to make a compliment about your work here.
Your Videos really change my view on maths and let me recognize the beauty in many topics.
Big Thanks
Thank you for not ignoring the cross product,as it is often done. (For example in School (in Germany (at least))).
This is very amazing video to help me to understand how vector and matrix work. Thank you so much for such a wonderful explanation!!!
Would be cool if you could explain the direction of cross products. The right hand rule is helpful but one of my favorite things about these videos is that you help make things intuitive rather than just rules.
It's possible to define (or, if you prefer, extend the definition of) cross product into n dimensions, where n ≥ 2. The result is a tensor (actually, a pseudotensor) of (tensor) rank n-2.
So only in n=3 dimensions, is the result a vector.
In n=2, it is a scalar - exactly the one you showed here;
in n=4, it's a tensor (square matrix);
in n=5, it's a rank-3 tensor (cubical matrix);
etc.
Fred
It only really needs enough components to define the plane the two vectors share, which is N choose 2 in N dimensions. 4D has 4 basis vectors, and 6 distinct planes connecting them. 2D only has 1 plane, so there's only 1 required component.
This is awesome..these videos are blowing my mind.
Well watching it the second time after actually taking the lecture in University and holy moly this completes it. Thank you so much
I didn't get at all why the antisimetryc property of the cross product until now. Your work is amazing and I promise you that when I start to earn more money, I will totally go to patreon to help funding this amazing content. Thanks for everything. I used to feel like an idiot because I was not able to process this things, until I got here and clarifies all. I love all your videos.
Even if not directly related, I gotta say this is giving me the right intuition to understand the concept behind covectors and cotangent spaces in differential geometry!
Im really greatfull to your team for doing extraordinary videos ,thank u.
O.o it's always pretty cool when a professor you've had for a class is quoted in a video.
Su labor es de alto valor social, su trabajo es excelente. Desde américa latina, agradezco de corazón lo que usted hace...
Your videos are a dream come true.
As a physics student who learned how to compute dot and cross products long before taking any linear algebra, I learned about the trig formulas as well, where u.v = (u)(v)cos(a) and u x v = (u)(v)sin(a) (using the right hand rule for direction). I've never actually taken a linear algebra course and I'm shocked by how much I'm already familiar with from physics classes. I'm literally watching this series because I need to understand rank 2 tensors and eigenvectors/eigenvalues(?) for physics. Having to apply mathematical tools years before formally learning about them is the nature of a physics degree.
7:16 lol, I always have put my unit vectors on the top row when computing cross products
it's the same
@@benjaminojeda8094 obviously, the notation is just rotated 90 degrees
You are a genius, I understand now. Very well done.
you are teaching skill is very awesome. I am learning how to teach from you!
amazing as usual!
wow it mesmerizing, outstanding. thank you for sharing. Keep it up.
great,insightful videos
I just had electromagnetism explained to me in high school, but ignoring all the matrix-related stuff, and was so confused as to why would a moving charged particle have a force applied perpendicularly to the magnetic field vector. This is truly enlightening!
Very insightful. Thank you.
But why is the direction of the new vector perpendicular at all? It makes no sense. I have no intuition for this and I hate just accepting it.
Kevin McInerney Its like a coin. The product points heads. So parallelogram is coin, and we define that as heads and say it points that way by the Right hand rule. Savvy?
It's because the cross product is a travesty. Go and look up "Geometric algebra", look at the geometric product and the bivector it creates. It's a much more intuitive way of looking at things.
@@1951split this comment needs to be sticky at top ... the cross product is seriously mind-bending non-generalizing ......
dot or scalar is "nicer"
as JH suggests .... Please look up bivector ... vectors are like lengths so intuitively cross product is like area ...
also if you don't like it .... don't call it perpendicular .... just remember that the "other product dot or scalar is zero" ..... like if a x b = c then both
a.c = b.c = 0 ..... THIS is the only perpendicularity you need .... and this you can understand geometrically as volume of box with two sides same in magnitude and DIRECTION (if you consider the three sides meeting at a corner here a,a,b or a,b,b) hence zero volume (just as cross gives you "c" as vector representation of area of parallelogram)
Thank you very much for your wonderful videos! Please, could you also create a lecture series on tensors?
explain very clearly , thanks!!
Seeing the quality of these linear algebra videos, I think there should definitely be a series on geometric algebra in the future! It's another beautiful way to understand vectors, and a great unification of complex numbers, quaternions, and vector geometry.
thankx....it was really helpful.... keep doing the good job
Fantastic....an eye opener
Great explanation thank you!
Definitely buying your merch one day.
His voice is peacful and calm like meditation, meditation with math.
Wow, I'm taking notes on these videos because I'm trying to learn physics, and I can't believe how many more connections I am making then when I only watched the videos for Essence of Calculus. Now that I am trying to paraphrase the visualizations and concepts into my own words, I'm remembering the concepts a lot more and I even ended up using the Feynman technique on accident.
It's pretty incredible how these videos can serve not only to increase one's interest in math, but can also teach complicated subjects like linear algebra through basic intuitions which makes the subject a whole lot easier to digest if you are either taking the course, or trying to learn it via some online method such as MIT OCW. And hey I can finally understand some of the high school math that was soooo, SOOOO, SOOOOO archaic and just seemed like mixing and matching numbers in some foreign pattern, like I can understand it all so much clearer now it's incredible, but I'm not surprised, this channel has the some of the best content for visualizing difficult on subjects and it's pretty awesome what Grant does.
I'm really enchanted!!!
looking forward to the next video!
You need to revolutionize education by hiring some people to build similar videos of conceptual visualization for many subjects. If you could do multivariable calculus for me that would be awesome!! :)
Bo Peng He has already made a multivariable calculus course. It is available for free at khanacademy.org
Douglas Espindola Wow, didn't know this guy (Grant Sanderson) did multivariable calculus at Khan. I've done some of the MIT multivariable calculus (MIT 18.02) videos on RUclips by Denis Auroux, but have a feeling the ones done by Grant are gonna be really good too. Added to my very long "to do" list!
Yeah, his course at Khan always help me when I'm stuck in some subject. It's a very intuitive course and it's conceptual visualizations are awesome!
Douglas Espindola Even more excited to do that course now.
bo peng fruitfullness?
I keep binge watching this, I love this series. Linear algebra has always been a huge stone in my shoe. Sure I managed to pass the course with good grades, but forgot everything right away because I never really understood it, and everytime it showed up, which in electrical engineering is A LOT, I started sweating cold. From now on everything is gonna become so much clearer, this is realli a life-changing video series. I kept it there waiting for a long time, but now that I started I can't stop watching it, it's too amazing!
This channel just not only helps in studies...it surely inspires
I am eating all this up. YUMMY !
This is the perfect prerequisite for calculating normals in my 3d renderer
Great! Now I can do electromagnetic fields. Thank you.
Now that I think about it: You should do a video about orthogonal bases and how to compute the coefficients! It not only belongs to the most basic concepts of linear algebra (imho) but would also yield further insight into that vector determinant formula for the cross product.
Great video
Jeez, I wonder if you're gonna talk in the next episode about transformation of the vector to skew-symteric matrix and simply multiplying matrix with vector to get a result vector of a cross product. After proper lecture about meaning of cross product we were shown this, well, trick at robotics course and It made life so easy^^
Anyway, I love your videos!
Just in case you were wondering which direction is positive in the 3d plane example at 6:08 then look at the arrows of the axes which represent the direction of positive. I was so focused on applying the rule to the point I wasn't able to see these arrows
Thank you!
Hey 3b1b you should really cover Geometric Algebra (Clifford Algebra). The wedge product is so much more intuitive. I really feel the cross product is unintuitive and holds back a great portion of useful physics because it basically interprets something that should be a plane as a vector. And it is in my opinion holding back progress in physics education. The answer to it is to use instead the wedge or outer product, the geometric product and bivectors. Cheers I hope people really look this up if they are confused!!
dawg the video on geometric algebra made me more confused
I tried to subscribe to you, and learned I already am.
I think this is the first time that's ever happened, for a RUclips commenter.
Just keep making more videos of these kind.
thank you so much
Hi, I'm not sure you still answer comments on older videos. However, I wonder if you've ever read about geometric algebra or exterior algebra, and the concept of a bivector being the dual of a vector that spans a perpendicular area. Similar ideas at play here and would be a very interesting video to watch! I feel like that subject is fraught with confusion, the kind your videos usually clear up
This question came up in my mind while learning about the parellopiped area. My teacher would find the area b/w the two vectors using cross product leaving me frustrated about why wouldn't we get a new vector instead of an area. Thanks for clearing the doibt
Thank you
I love your work!!! Keep it up :)
P.s I am a programmer and I would be super interested in your series on Math and Programming around how programming you can learn and improve both math and problem solving ... But I understand if you only focus on Mathematics :)
Learning MATLAB helped me so much with linear algebra, just because it’s so much easier to play with numbers
good work, i thankyou
I just read somewhere that "If a Matrix has only one row or only one column, it is called a vector". Which means vectors are actually transformations which I think is a lot easier to think about although I am not sure my reasonning is right.
thanks for all your superb videos. Regarding dot and cross product, i would like to suggest that you make a video about Geometric Algebra, and explain the geometric product, which contains the inner product and the wedge (outer) product at the same time. The wedge product is quite similar to the cross product, but it seems more 'natural'.
Watching this video really helped me to mentally solidify the concepts. Thank you for creating these videos and keep up the great work!
Really enjoying this series. This one had me really confused though starting with the 2D logic and then saying, "But that's not the cross product." By the end I thought maybe cross product was a different thing in 2D (an area) and 3D (a vector). Had to go back and watch again the next day to realise the 2D bit was just setup to describe what the cross product actually is, and that it only (?) applies to 3D vectors, not 2, 1 or > 3.
oh bless your soul
Thank u man
Awesome video!
No, his formula is correct. The ĵ term in this setup will have the opposite sign of the other terms, which is why the formula seems flipped.
@@MuffinsAPlenty Oh I see, yes. You are right....Apologies.
5:43 & 1:13
in fact, you can pick any of the possible two as "the positive orientation of 3D space", which one is more convenient for you- it's absolutely non-mandatory (or maybe it is habitual or ignorant/overtaught or unaware or whatever) to "follow the right-handed helix" to have a positive v × w
The way you make these concepts intuitive is out of this world. You really put every single course on linear/non-linear algabra I've had to shame. This is precedent for teaching done right.
What music do you use for your intro?
I don't know where to make a request but can you pleasee make a series explaining financial math, such as correlation, volatility etc. Your visuals and way of explaining it is so helpful to understand math concepts and what they mean in real world.
Am I right in thinking that most people watching this are probably gonna watch the next one?
Edit: just checked viewcount, and I am. Gotta love when 39/40 of people want to go deeper.
Hehe.... That last sentence tho😏😏
its interesting, im learning this in class now, and its being taught to me a different way. its essentially the same thing, but the matrix is flipped 90 degrees counter clockwise in this video. so if you have the vector and you put it in the matrix as
[i j k]
[1 2 3]
[4 5 6]
and get out 3 2x2 matricies like
[2 3] [1 3] [1 2]
[5 6] i - [4 6] j + [4 5] k
then find the crossy thing like (2x6)-(5x3)=-3i and do thqt for the others so it would be 6j and -3k, so then the vector perpendicular to both of those would be and if you do the dot product of those, you get 0, confirming it
• and you get -3+12-9=0 and
• = -12+30-18=0
therefore proving it is perpendicular