 Welcome back to our lecture series about the textbook linear algebra done openly. As usual, I'm your professor today, Dr. Andrew Misaline. We are in section 5.4 today entitled Cross Products. Now Cross Products is a type of multiplication between vectors. But one little caveat I have to mention here, not a little caveat, it's sort of a big one here. This Cross Product is only going to be fine for the vector space F3. That is, we have to have exactly three coordinates. So our vector U, we can write as U1, U2, U3. And our vector V, we can write as V1, V2, and V3. Now we're not specifying what the scalars have to be. They just have to come from a field. A little bit later in this lecture, we will restrict to real numbers. But for the moment, we can allow any scalar right here. And so when it comes to two vectors, U and V, we define their cross product, which the word cross here just represents the notation to represent the multiplication. Because we're starting to see there's lots of different multiplications of vectors. There's the dot product, which we take two vectors, U, U dot V. And then this gives us a scalar, right? And as such, this is sometimes called the scalar product or the inner product, right? We had the tensor product before. We had things like U tensor V. So you draw a circle around the X. This would output a matrix. And as such, the tensor product sometimes called the matrix product or as we call it the outer product, right? We talk about the location of the transpose symbol. The cross product is a product of vectors and it actually produces a vector. If you take U times V, this will give you a vector when you're done. So the dot product gives you a scalar. The tensor product gives you a matrix and the cross product gives you a vector. And so sometimes it's called the vector product. So what that vector looks like is the following. Combining the coordinates of U and V, the first entry of U times V will be the vector U2 V3 minus U3 V2. You'll notice this is a product of entries from U and V, the second and third together. Do pay attention to the minus sign and you'll notice that the first entry is nowhere appearing here. So no U1 or V1 there. For the second entry of the cross product, you get U3 times V1, subtract from that U1 V3. So again, it only involves U1 V1, U3 V3 and you take the products of each other and you're just going to subtract the product and the second entry is not used at all. And then lastly, for the third entry of this vector, you get U1 V2 minus U2 V1. So you're combining again products between the U's and the V's. You use the first and the second coordinate but not the third coordinate. So you don't actually use the third coordinate to calculate the third coordinate of the cross product. That's true for the other ones. But then the issue is about, okay, if I take U1 times V2, that's not, you'll remember to do one and two and two and one but the presence of the minus sign makes a difference here, right? Which one do you subtract? If you just try to memorize this out of context, it kind of feels like alphabet soup and you're going to have a hard time trying to remember it all. And the main reason we postponed this discussion of cross products until the end of chapter five is so that we can make a connection to determinants because after we've learned how to calculate determinants, turns out it's actually quite straightforward how to calculate cross products. Let me actually go back to the original formula. If you look at the following determinants, two by two determinants, if you take the matrix U2 V2, U3 V3, you calculate it's determinant. So you take that cross like that, then you get exactly this formula, U2 V3 minus U3 V2. And then if you take the second one, you're going to take U1 V1, U3 V3, if you take its determinant, you end up with a U1 V3 minus U3 V1, which is this formula backwards, but a way we're going to stick a negative sign to correct that, the presence of that negative sign will become even more clear in just a second. And then for the last one, you'll take the determinant U1 V1, U2 V2, which when you take the cross product there, you get, well, the cross product, okay, there you go, U1 V2 minus U2 V1, like so. And so these formulas you see in the original formula are just determinants of two by two matrices, but why the minus sign right here? And why these determinants, why are we choosing these three? Well, if you take a step down here, I can even make it a little bit more enlightening, more enlighten what's going on here. Consider the matrix three by three matrix you see here, which has the entries E1, U1 V1, E2, U2 V2, and E3, U3 V3. Now it's kind of hard to see here, but these here are bolded, these are vectors. So E1 here, this is the vector 1, 0, 0, and E2 is the vector 0, 1, 0. And lastly, E3 is the vector 0, 0, 0, 0, 0. Sometimes these vectors like in a calculus setting or in like a physics setting, they might call this vector I, this vector J, and this vector here K. In this lecture series, we have mostly ignoring those labels because they're kind of specific to three dimensions while we worked over multiple dimensions in linear algebra. But again, with the cross product, we're kind of stuck in three dimensions because of this picture we have right here. So with this matrix, if you were to cofactor expand this matrix along the first column, you would end up with E1 times by this determinant right here. That is, you would get the minor U2 V2, U3 V3, which is exactly what you see right here. Or if you did the second one, you get E2 times, because you take out this column or this row right here, you get U1 V1, U3 V3, that's exactly what you have right here. Now, of course, the first entry was 1, 1. This is the 2, 1 entry by rules of cofactor expansion. This first one was a positive, this one's a negative, and then the 3, 1 position should have a positive coefficient. And that's exactly what you see right here. You see this plus minus plus alternation going on. All right? And so that kind of explains why there's a minus sign there. And then lastly, if you take away the third row from the cofactor expansion, of course you take away the column too, you're gonna get E3 times the minor. And so in terms of determinants, it kind of makes sense where this formula comes from. Now, I should try to reassure you here that although the second and third columns of this matrix are scalars, the first column here actually consists of vectors. So as of yet, we've only put scalars inside of matrices. I'm now allowing for the possibility of putting vectors or maybe matrices as entries of matrices. Notions like of matrix addition, matrix multiplication, determinants can also make sense in this situation when we allow for non-scalar entries to be put in here. And so this gives us a situation where we might actually care to have a vector entry in part of the matrix. And this cross product is none other than just this determinant. So let's see an example of such a thing, right? Take the vector U to be one, two, negative two and V to be the vector three, zero, one. Then the cross product of these things we can calculate as the determinant E1, E2, E3, one, two, negative two. So we have a column for U and we have a column for V three, zero, one, like so. Now I wanna mention that one thing to remember here is that when it comes to the determinant, if you have a matrix, the determinant of the transpose is the same thing that is taking a determinant, or taking a transfer doesn't affect the determinant. And so when people work with this, sometimes instead of writing them as columns, people prefer to write this as rows. So you have the row vectors E1, E2, E3 and then you write U as a row, one, two, negative two and you write V as a row, three, zero, one. And then like I said, these E1, E2, E3s are often written as like these other unit vectors I, J and K. So like in a physics or a calculus textbook, you might see the determinant or the cross product defined in this manner right here. Again, that's totally equivalent. So don't feel like you have to be one or the other. As I usually like to think of vectors as column vectors, I'm gonna write them as columns in this determinant, but heck, I wrote it as a row vector right there. So you can do either one. So if we cofactor expand along the first column, let's see if we take out the first row and first column like that, you can see your minor that's left behind, that'll give us two, zero, negative two, one times E1. And then for the next one, since you're gonna subtract it, second row, second column, you'll get the minor one, three, negative two, one. One, three, negative two and one, E2. And then lastly, if you take off the third row first column, you get the minor one, three, two and zero, like so. And so if we calculate these two by two determinants for the first one, you'll get two, one minus zero times negative two, that'll give you a two as your first entry. For the next one, you're gonna get one, one minus, minus a negative six, that actually gives us a seven, but don't forget the negative sign that's in front there, you actually get a negative seven for your second entry. And then for the last one, you take one, zero, minus three, two, that's gonna give you a negative six, which we'll record right here. Let me write that six a little bit better. So that gives us our cross product there, two, negative seven, negative six. And so you can use this determinant calculation to help you remember the cross product. Now, admittedly, in terms of computational speed, it'll be faster just to plug those six numbers into the original formula. So if you were to put this in like a computer program, just memorize the formula for the program and plug them in there. But as a student where it's just like, just memorizing random symbols, it seems crazy. But if you see it in the context of the determinant, it's hard not to remember these formulas right here.