Cross products | Chapter 10, Essence of linear algebra

This covers the main geometric intuition behind the 2d and 3d cross products.
An equally valuable form of support is to simply share some of the videos.

*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.

Future series like this are funded by the community, through Patreon, where supporters get early access as the series is being produced.

Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Vietnamese: @ngvutuan2811

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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).

Various social media stuffs:

[patrickjmt]

These videos are just so damn good. Kudos to you Grant, keep up the great work!

[Cerealbox]

"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.

[noahmccollum-gahley]

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.

[sasjadevries]

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.

[abhishekshah]

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.

[alejrandom]

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

[twilightknight]

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!

[brglackinify]

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.

[kjekelle]

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

[Pivotcreator]

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

[carlosalberto]

"That was technically not the cross product"
*angry pi

[Sweetyfragolina]

These videos are getting weirdly addicting

[dorol]

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!

[estefaniakiara-elizabeth]

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!

[MrZach]

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.

[Muuip]

A true role model for what teaching online is becoming.
Bravo and thank you!

[philipgiacalone]

This whole series is brilliant! Thank you so much for putting this excellent material together. It was clearly a lot of work.

[nathandaix]

This is one of my favorite series! I love the visualization I get from these videos. Great work and keep on making awesome content!

[oxupylg]

I almost done translating all of this series to Hebrew CC. Please approve this CC as well.

Yours truly :-)

[justalettertooshor]

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.