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Cross products | Essence of linear algebra, Chapter 10

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This covers the main geometric intuition behind the 2d and 3d cross products. *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: http://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. http://3b1b.co/support ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: https://goo.gl/WmnCQZ Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3Blue1Brown Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown Reddit: https://www.reddit.com/r/3Blue1Brown
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Text Comments (202)
Parth Sakariya (1 day ago)
Just keep making more videos of these kind.
I would watch your next video but Ive noticed I often dont get it. I understand your vids after Ive already taken a class on it and everything begins to make sense so in a way your vids are what connect the messy knowledge thats given to us.
Hui Shang (9 days ago)
This is very amazing video to help me to understand how vector and matrix work. Thank you so much for such a wonderful explanation!!!
EPISTEMOLOGIST (12 days ago)
at 2:24 i think you meant chapter 6. great video tho
퍼플유아이 (15 days ago)
Omg bts on advertisement♡
Dnyaneshwar (15 days ago)
Not reachable to student need
sathiya seelan (17 days ago)
pls explain about the eigen vector eigen value
Shamim Yeasir Saion (18 days ago)
Zach Schmitz (19 days ago)
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.
Piyush Ranjan (21 days ago)
https://youtu.be/eu6i7WJeinw?t=458 The determinant will have a negative sign before j' in the computation.
dd ff (21 days ago)
Song at the start of the video?
Ajeet Kumar (23 days ago)
Wow sir ji...
Hai Nguyen Trong (1 month ago)
The cross products determines the spatial orientation shift of a product function into a volume function but its end goal cannot implement its overall temporal transformation when it faces to a jump gap shift spacial gap. In these terms we have to find a new solution to overcome the spatial gap in order to restrict the evolution of the factual 4 D transformations or 3D compound linear transformations if not the progressive overall linear transformations of our vectors cannot answer to the spatial cohérence in progressive time shifting or origin alternate base shift of our completion successives vectors. If we do not cure this problem we cannot precisely correct the spatial base origins shift of the spatial gap encounters in the split cross products of our research. Our research has to meet the end results of an évolutive non linear cross products split functions to answer to the intricacies of an non define global functions to its core.
CubeCraftGalaxy (1 month ago)
What software do you use to make these videos?
Kira Sama (1 month ago)
At 7:30, shouldn't it be *j(v1.w3 - w1.v3)* ?
David Bang (2 months ago)
Without this lecture, people only know a determinant is just a number and cross product is just a tedious process of producing another number.
Varun Kumar (2 months ago)
Is matrix product and victor product are different???? we can't product M(1x3) and M(1x3). May i know how to product M(1x3) and M(1x3) if they are vectors.
Emiliano S (2 months ago)
Please! Somebody tell me the link to the video of graph duality.
zvxcvxcz (3 months ago)
O.o it's always pretty cool when a professor you've had for a class is quoted in a video.
Paige G (3 months ago)
Wtf I'm so lost I barely got an A in pre calculus in high school...
Semin Park (3 months ago)
Thanks for making these videos. They are very enlightening! I have one question though. I thought that the cross product is not well defined in dimensions other than R^3. I googled a bit and found that there are several ''analogues'' of the cross product which in 2D could be the one that is introduced here(taking two vectors as inputs), or you could also find the orthogonal vector of the input vector (in which case the input variable would be just *one* vector, not two). My question is, since the cross product is not well defined in 2D, wouldn't 3b1b have to make it clear that he's defining something that is not a standard way of calculating the cross product in R^2?
Peter Amobi (3 months ago)
Does the v and w signify anything or it can be any letter... is v always the longer vector
John Li (3 months ago)
so in 4D, there are two linearly independent "3rd" vector, what do you do???? how do you define cross product as magnitude = area of the plane and direction normal to the plane??
Joseph Groves (3 months ago)
3:30 woah. That makes sense now
Lim Ri Bing (3 months ago)
explain very clearly , thanks!!
CAILZZZ (3 months ago)
7:38 shots fired !!
Riddhi Chatterjee (3 months ago)
Saul Moreno Gomez (4 months ago)
What kind of tool do you use to make this excellent videos ?
dank apologist (4 months ago)
Oh man I'm in mensa and I feel stupid
Eshika Lad (4 months ago)
Can you make playlist for vector calculus?
Samia Zaman (5 months ago)
Molly Pete (5 months ago)
Your videos are a dream come true.
Omar Maswadeh (5 months ago)
What the!!!!! Please do videos on tensors and vector calculus and analysis and groups and topology and graph theory and everything in mathematics please, I adore your simplifications 😭
Alex Gil (5 months ago)
Su labor es de alto valor social, su trabajo es excelente. Desde américa latina, agradezco de corazón lo que usted hace...
Csanad Horvath (5 months ago)
I wish my teachers were like this!
ableite (5 months ago)
is there a video explaining div curl and grad?
Graham Lea (5 months ago)
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.
hj labs (6 months ago)
0:07:35 mistake.... -j should be there...
Cristian Del Toro (6 months ago)
Those Pi's are so expressive
GZ (6 months ago)
A thumb up to this video is [0, 0, 1]
Ahmed boudi (6 months ago)
thank you so much
Manuel Biwer (7 months ago)
Can you use this trick at 7:16 in 4D
Elijah Hua (7 months ago)
Seriously, why perpendicular? No one explain it before, they just say, oh it is perpendicular,
Elijah Hua (7 months ago)
Why perpendicular!
Gurpartap singh (7 months ago)
thankx....it was really helpful.... keep doing the good job
Chandarshi dronamraju (7 months ago)
Please explain complex analysis
D Singh (7 months ago)
Great work 👍 May I know what software do you use for all the presentation animation. Thanks!
M grisez063 (5 months ago)
He does not use software, his animations are directly coded by himself
MrShaylois (7 months ago)
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.
Caridorc Tergilti (7 months ago)
You are a genius, I understand now. Very well done.
lice1st (8 months ago)
In Russian math school we call dot product a "scalar product" and a cross product a "vector product". And I think this is very intuitive naming
I I (9 days ago)
lice1st scalar product usually refers to any product which is scalar and a vector product for any product which is a vector. So it's not tied to only dot and cross products. For example af(x) where a is a scalar constant and f is a function, we get a scalar product.
Fi La (5 months ago)
lice1st not only in Russian school, of course. It doesn't really give you any more intuition about what they mean, it only reminds you of what each of them produces - a vector or a scalar.
Thomas Ip (8 months ago)
If the magnitude of the normal vector is the area of the parallelogram formed by v and w, how do I find the area using the determinant? (the matrix from v and w is non-square (when v & w is 3d) so there is no determinant)
12388696 (8 months ago)
You downgraded a higher education topic to an elementary one.
Michael K (8 months ago)
@3Blue1Brown, do you even realize how amazing these videos are?
D Wright (8 months ago)
j-hat should be minus at 7:30 in this video. Sorry to nag, but the video is so great, I want it to be perfect!!!
Atharva #breakthrough (8 months ago)
great videos. I have 2 notes regarding 2.50 : cross products are not defined for R^2 cross product is a vector, not a scalar (Determinant)
eel0102 (8 months ago)
From Calc 3, I thought the cross product was a vector, and the magnitude of the cross product was the area of the parallelogram. Is this totally different? These are beautiful videos by the way:)
K1naku5ana3R1ka (9 months ago)
I learned that cross-as-a-determinant process with i-j-k on top, but I suppose it works out the same.
Andrew Church (10 months ago)
why does cross product only work in 3 and 7 dimensions?
rrr00bb (10 months ago)
Geometric Algebra - Wedge Product
Nathan Rogers (10 months ago)
I love your material, immensely. Don't get me wrong, you're incredible at explaining things. But your pi dudes are irritating to look at.
Z Feldberg (10 months ago)
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.
Philip Giacalone (11 months ago)
This whole series is brilliant! Thank you so much for putting this excellent material together. It was clearly a lot of work.
Johnson Joseph (11 months ago)
This channel just not only helps in studies...it surely inspires
Kevin McInerney (11 months ago)
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.
Pavel Shliaha (1 year ago)
could you please make a video explaining why matrices need to have certain dimensions to be multipliable, i.e. mxn and nxp
Maiyuan Wester (1 year ago)
Siu Harry (1 year ago)
I would like to ask a question. The column of V does not match with the row of W. So how come there could be cross product?
akshay tamhankar (1 year ago)
right hand screw can be used to determine sign of cross product
Rico Wang (1 year ago)
anyone can tell if the determinant calculation on 4:10 is wrong?
Rico Wang (1 year ago)
It's correct, sorry.
Alexander N. Benner (1 year ago)
Thank you for not ignoring the cross product,as it is often done. (For example in School (in Germany (at least))).
Im really greatfull to your team for doing extraordinary videos ,thank u.
vadimuha (1 year ago)
Used right hand rule for perpendicular example, broke hand
P equals NP (1 year ago)
What program are u using
the5chronicles (1 year ago)
is this similar to the wedge product?
Nina Calgary (1 year ago)
What is the name of the intro music?
Sarath Sivaprasad (1 year ago)
great,insightful videos
Brendan Glackin (1 year ago)
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.
Carrot Cake (1 year ago)
You should join Khan Academy as a full time teacher.
HPP (1 year ago)
Thank you very much for your wonderful videos! Please, could you also create a lecture series on tensors?
Caio Oliveira (1 year ago)
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.
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!
Sean Dafny (1 year ago)
I am eating all this up. YUMMY !
Caterina Puca (1 year ago)
These videos are getting weirdly addicting
Ali Barkhordarian (1 year ago)
So is the cross product only for three dimensional vectors?
Suzuki Kenta (1 year ago)
Cross product seems to be applicable in higher dimensions. Generally it takes n-1 vectors in n dim and spits out a vector orthogonal to those n-1 vectors. You can check this argument, http://math.stackexchange.com/questions/185991/is-the-vector-cross-product-only-defined-for-3d
Ali Barkhordarian (1 year ago)
Ey jomike thanks.
Ey jomike (1 year ago)
Yes it is
Brock Fettes (1 year ago)
These videos are amazing, but the right-hand rule really bugs me. If you use the left hand, with x along the middle finger and y along the first finger then your thumb naturally points along z. This means the fingers are labelled in order x, y, z. Using your right hand, not only are your fingers labelled y, x, z but you also have to bend it over backwards to get it into the right orientation.
Aziza Zhanabatyrova (1 year ago)
Hi, it'd be nice if you could give some examples of applications in real life
Sanji Vinsmoke (1 year ago)
The Extremist (1 year ago)
Armando José (1 year ago)
Thanks for your video, your channel is amazing and very illustrative, I really learn a lot about maths and things about it that I never thinked about so deeply. About the cross product, can I understand it as the normal vector of the parallelogram?
Armando José (1 year ago)
Thanks! This illustrative version of the cross product is much more easy to comprehend
3Blue1Brown (1 year ago)
Yup! In 3d, the cross product of two vectors is always perpendicular to each one, which also means it is perpendicular to the parallelogram spanned by the two of them.
Muuip (1 year ago)
A true role model for what teaching online is becoming. Bravo and thank you!
Mickey Zacho (1 year ago)
Shouldn't it be i(v2w3-v3w2)-j(v3w1-v1w3)+k(v1w2-v2w1)? Note the "-j(v3w1-v1w3)" instead of adding this?
Eelke Johnson (1 year ago)
You are right i was also wondering but in the video this is false
FinsaidHi (1 year ago)
no it would be i(v2w3-v3w2)- j(v1w3-v3w1)+k(v1w2-v2w1) but he's taken out a common factor of - 1 so it is i(v2w3-v3w2)- j(-(v3w1-v1w3))+k(v1w2-v2w1) which simplifies correctly
Koen Kwakkenbos (1 year ago)
I was wondering this as well :)
Abhishek shah (1 year ago)
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.
Samuel Andrade (1 year ago)
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
Noah'sKnowledgeCenter (2 months ago)
Samuel Andrade you could translate the video ask someone to translate the video for you.
The Flagged Dragon (1 year ago)
Can you make some videos on exterior algebra?
Don Grille (1 year ago)
man this channel is so awesome. math is so awesome
oldcowbb (1 year ago)
i think the right hand grip rule is easier to remember than the three finger right hand rule, it's very easy to mess up which finger is which vector
sasja de vries. (1 year ago)
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.
purewaterruler (1 year ago)
The only problem is that you introduced the cross product as the area of the parallelogram instead of ad the vector
Carlos Montoya (1 year ago)
"That was technically not the cross product" *angry pi
StarchyPancakes (2 years ago)
My thought process at the end of the video: "Wait, if this is true, that would that mean that you could take any trio of 3 dimensional vectors, change one of them to the unit vector, and then the determinate would be the a.... Oh. Holy Shit."
duc unknown (2 years ago)
when you calculate vector v cross vector w using that notational trick at 7:09, i notice that in vector j's scalar the way the components were added were in reverse compare to vector i's scalar and vector k's scalar. Was that because when you look at the x and z axes, the x-axis is on the left of the z-axis, so when you cross 2 vectors in the xz-plane, the usual negative orientation would be the positive orientation in the xz-plane? p.s. i am sorry if my question is a bit long and confusing for you, i am learning linear algebra in a different language so i can't explain it in English well.
Fauxmar (1 year ago)
If you look back to his video on determinants, he mentions the algorithm for computing the determinant of a 3x3 matrix. As you go down a row or column to use each index as a scalar (i, j, k in this case), the sign alternates. Here's an example of the signs in a 3x3 matrix + - + - + - + - + So in this case, he distributed the j's negative which is why it is the opposite from i's and k's 2x2 determinants. That's the numerical explanation anyway.
N (2 years ago)
This is one of my favorite series! I love the visualization I get from these videos. Great work and keep on making awesome content!

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