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
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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.
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.
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.
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?
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.
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.
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)
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.
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!
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
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.
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?
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.
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.
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."
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.
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.
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