Cross products | Essence of linear algebra, Chapter 10

Cross products | Essence of linear algebra, Chapter 10


Last video, I’ve talked about the dot product. Showing both the standard introduction to
the topic, as well as a deeper view of how it relates
to linear transformations. I’d like to do the same thing for cross products, which also have a standard introduction along with a deeper understanding in the light
of linear transformations. But this time I am
dividing it into two separate videos. Here i’ll try to hit the main points that students are usually shown about
the cross product. And in the next video, I’ll be showing a view which is less commonly
taught, but really satisfying when you learn it. We’ll start in two dimensions. If you have two vectors v̅ and w̅, think about the parallelogram that they span
out What i mean by that is, that if you take a copy of v̅ and move its tail to the tip of w̅, and you take a copy of w̅ And move its tail to the tip of v̅, the four vectors now on the screen enclose
a certain parallelogram. The cross product of v̅ and w̅, written with the X-shaped multiplication symbol, is the area of this parallelogram. Well, almost. We also need to consider orientation. Basically, if v̅ is on the right of w̅, then v̅×w̅ is positive and equal to the area of the parallelogram. But if v̅ is on the left of
w̅, then the cross product is negative, namely the negative area of that parallelogram. Notice this means that order matters. If you swapped v̅ and w̅ instead taking w̅×v̅, the cross product would become the negative of whatever it was before. The way I always remember the ordering here is that when you take the cross product of the two basis vectors in order, î×ĵ, the results should be positive. In fact, the order of your basis vectors is what defines orientation so since î is on the right of ĵ, I remember that v̅×w̅ has to be positive whenever v̅ is on the right of w̅. So, for example with the vector shown here, I’ll just tell you that the area of that parallelogram is 7. And since v̅ is on the left of w̅, the cross product should be negative so v̅×w̅ is -7. But of course you want to be able to compute this without someone telling you the area. This is where the determinant comes
in. So, if you didn’t see Chapter 5 of this series, where I talk about the determinant now would be a really good time to go take a look. Even if you did see it, but it was a while ago. I’d recommend taking another look just to make sure those ideas are fresh in
your mind. For the 2-D cross-product v̅×w̅, what you do is you write the coordinates of v̅ as the first column of the matrix and you take the coordinates of w̅ and make them the second column then you just compute the determinant. This is because a matrix whose columns represent v̅ and w̅ corresponds with a linear transformation that moves the basis vectors î and ĵ to v̅ and w̅. The determinant is all about measuring how areas change due to a transformation. And the prototypical area that we look at is the unit square resting on î and ĵ. After the transformation, that square gets turned into the parallelogram that we care about. So the determinant which generally measures the factor by which areas are changed, gives the area of this parallelogram; since it evolved from a square that started with area 1. What’s more if v̅ is on the left of w̅,
it means that orientation was flipped during that transformation, which is what it means for the determinant to be negative. As an example let’s say v̅ has coordinates negative (-3,1) and w̅ has coordinates (2,1). The determinant of the matrix with those coordinates as columns is (-3·1) – (2·1), which is -5. So evidently the area of the parallelogram we define is 5 and since v̅ is on the left of w̅, it should make sense that this value is negative. As with any new operation you learn I’d recommend playing around with this notion of it in your head just to get kind of an intuitive feel for what the cross product is all about. For example you might notice that when two vectors are perpendicular or at least close to being perpendicular their cross product is larger than it would be if they were pointing in very similar directions. Because the area of that parallelogram is larger when the sides are closer to being perpendicular. Something else you might notice is that if you were to scale up one of those vectors, perhaps multiplying v̅ by three then the area of that parallelogram is also scaled up by a factor of three. So what this means for the operation is that 3v̅×w̅ will be exactly three times the value of v̅×w̅ . Now, even though all of this is a perfectly fine mathematical operation what i just described is technically not the cross-product. The true cross product is something that combines two different 3D vectors to get a new 3D vector. Just as
before, we’re still going to consider the parallelogram defined by the two vectors that were crossing together. And the area of this parallelogram is still going to play a big role. To be concrete let’s say that the area is 2.5 for the vectors shown here but as I said the cross product is not a number it’s a vector. This new vector’s length will be the area of that parallelogram which in this case is 2.5. And the direction of that new vector is going to be perpendicular to the parallelogram. But which way!, right? I mean there are two possible vectors with length 2.5 that are perpendicular to a given
plane. This is where the right hand rule comes in. Put the fore finger of your right hand in the direction of v̅ then stick out your middle finger in the direction of w̅. Then when you point up your thumb, that’s
the direction of the cross product. For example let’s say that v̅ was a vector with length 2 pointing straight up in the Z direction and w̅ is a vector with length 2 pointing in the pure Y direction. The parallelogram that they define in this simple example is actually a square, since they’re perpendicular and have the same length. And the area of that square is 4. So their cross product should be a vector with length 4. Using the right hand rule, their cross product should point in
the negative X direction. So the cross product of these two vectors is -4·î. For more general computations, there is a formula that you could memorize if you wanted but it’s common and easier to instead remember a certain process involving the 3D determinant. Now, this process looks truly strange at first. You write down a 3D matrix where the second and third columns contain the coordinates of v̅ and w̅. But for that first column you write the basis vectors î, ĵ and k̂. Then you compute the determinant of this matrix. The silliness is probably clear here. What on earth does it mean to put in a vector as the entry of a matrix? Students are often told that this is just a notational trick. When you carry out the computations as if î, ĵ and k̂ were numbers, then you get some linear combination of those basis vectors. And the vector defined by that linear combination, students are told to just believe, is the unique vector perpendicular to v̅ and w̅ whose magnitude is the area of the appropriate parallelogram and whose direction obeys the right hand rule. And, sure!. In some sense this is just a notational trick. But there is a reason for doing in. It’s not just a coincidence that the determinant is once again important. And putting the basis vectors in those slots is not just a random thing to do. To understand where all of this comes from it helps to use the idea of duality that I introduced in the last video. This concept is a little bit heavy though, so I’m putting it in a separate follow-on video for any of you who are curious to learn more. Arguably it falls outside the essence of linear algebra. The important part here is to know what that cross product vector geometrically represents. So if you want to skip that next video, feel free. But for those of you who are willing to go a bit deeper and who are curious about the connection between this computation and the underlying geometry, the ideas that I will talk about in the next video or just a really elegant piece of math.

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  1. Su labor es de alto valor social, su trabajo es excelente. Desde américa latina, agradezco de corazón lo que usted hace…

  2. 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 😭

  3. 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??

  4. 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?

  5. Is matrix product and victor product are different???? we can't product M(1×3) and M(1×3). May i know how to product M(1×3) and M(1×3) if they are vectors.

  6. Without this lecture, people only know a determinant is just a number and cross product is just a tedious process of producing another number.

  7. 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.

  8. 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.

  9. ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤😉😉😉😉😉

  10. This is very amazing video to help me to understand how vector and matrix work. Thank you so much for such a wonderful explanation!!!

  11. i don't understand the i/j and v/w direction thing.. is the v and w th conventional notation used for any two vectors? How would use designate one v and one w when given a practical application problem?

  12. Since V x W = – W x V, for those having multivariable calculus, in particular having to check if a vector field is conservative or not, we use the cross product with the derivatives and the components of the field. V x W is non abelian, that is the order does matter, but in the case of finding a conservative vector field we only look for the zero vector (in order to prove it's conservativity). And if V x W != 0, the inverted order being – ( V x W ) will not be zero either.

    So in the case of finding conservative vector fields it doesn't matter if one confuses the order of which to cross-multiplicate on (Since we look for the zero vector, and õ = -õ, õ being the zero vector).

  13. it makes a lot more sense to use Geometric Algebra. x=e1, y=e2, i = (e1 e2) = -(e2 e1), (e1 e1) = 1, (e2 e2 ) = 1. (a b) = (a1 e1 + a2 e2)(b1 e1 + b2 e2) = ((a1 b1 + a2 b2) + (a1 b2 – a2 b1)(e1 e2) = (a . b) + (a ^ b) = (a . b) + i(a X b) = |a| |b| cos(theta) + i |a| |b| sin(theta) …. the rules about how the unit basis vectors multiply (ie: x = e1, y = e2) will generate all of 2D, 3D, 4D,… math. and there is almost nothing to memorize. it also makes all handedness conventions and row vs column conventions disappear. the standard notation is a big mess. with GA, the signs tell YOU what the orientation/handedness is, and your choice of (e1 e2) are pretty arbitrary as long as they are orthogonal and same unit length.

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

    Yours truly 🙂

  15. 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!

  16. I just wanted to know about this because I've read about cross vector products in a meme… And I think it's going to be my next topic in math because the animation looks like our last topic 😂

  17. 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.

  18. Just use right hand rule. It's faster than using determinant to know whether it's positive or negative direction.
    If your thumb points downward, it's negative.
    …..upward, it's positive.

  19. Better to use the cork-screw rule to get the direction of the cross product. Which vector is left of the other one doesn't work – a rotation of the common plane will change which vector is the one that's left of which without changing the cross product. The right hand rule is too difficult to apply when the vectors are not close to orthogonal.

  20. I think there is an error in the formula for the determinate at the end. The j-hat term should be negative shouldn't it?

  21. You got me confused. We got used to put ‘i’, ‘j’ and ‘k’ on top of the matrix that they are in a row. Here you kinda used different way. That’s why I didn’t know what was going on since I started watching the series of linear algebra videos. Gotta review them now.

  22. If you don’t have the mind blowing animation at your disposal, just draw a line on the x axis from one corner of the parallelogram to the other, and two more lines from the top corner to the x axis and the bottom corner to the x axis, and you will see that you have four triangles that can easily be shown to add up to the same thing as the determinant.

  23. And if you have a TI Nspire graphing calculator (and you're allowed to use it), you can just use the dot/cross Product function. You can see how here: https://www.youtube.com/watch?v=FETb-8_bupM .

  24. But the fact that the cross prpduct is negative here as opposed to positove is just an arbitrary,co vention right? It makes as much sense for the cross product of 2 in z and 2 in y to be positive 4..its just convention..which makes it confusing that they don't explicitly clarify that…

  25. Like always, thanks for the awesome video!
    In case anyone was confused about how he got to the cofactor equation, I asked this question on stack exchange:
    https://math.stackexchange.com/questions/3284365/cross-products-and-determinants-geometrically
    I hope it helps!

  26. sir please a video on vector triple product .
    because it is very amazing that three cross producted vector can be related to projection of one vector on other two …

  27. 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!

  28. A lot of this seems similar to what I learned when I studied geometric/Clifford algebra. Is that an approach to linear algebra that's more closer to the transformational approach you describe?

  29. It is said that the determinant gives the magnitude of the vector of cross product. Having two vectors in 3d space and doing cross product if means taking vector perpendicular to this parallelogram given by these vectors then how do i find magnitude using determinant from 2 vectors in 3d space.

  30. Bruh you said cross product is area which means result of a cross b is scalar but cross product gives you vector not scalar.That ''well almost'' is kinda misleading.Magnitude of cross product is area of parallelogram.Let me give you proof.Think of 2 vector and angle between them can make a parallelogram.For parallelogram we need height multiplied my breath.And for height we can use trigonometry's sin function multiplied by some height vector and then we multiply it with breath vector that I told you to think of.then you will get that area.There are another way which is shadow.You use torch from – to + position horizontal and do projection on vector.Sorry for being sloppy.If I have software the I will also do video on that.

  31. I don’t understand the right hand rule. I don’t see how this video visualizes it or makes it more intuitive. Can someone record a video with a real hand to visualize this?

  32. When you show the determinant of the 3×3 matrix, the j(hat) term should have been negative. The resulting equation from the determinant of a 3×3 matrix is always i(hat)-j(hat)+k(hat).

  33. If i remember correct, from Biot-Savart law on Magnetic Fields, theres one more definition for cross product:
    G = U x W = |U| * |W| * sinθ * r , whereas θ is the angle between U and W , and r is the perpendicular (to UW plane) vector and has length 1

  34. 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

  35. 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!

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