 We're going to take a look at the final aspect of vector algebra that we need in this course. And that is specifically another kind of vector multiplication known as either the vector product or the cross product. The key ideas that we will learn in this section of the course are as follows. We will revisit the idea that vectors have two kinds of multiplication, only one of which we have seen up until now. The one we've already seen is the scalar product or dot product, which takes as an input two vectors and it returns a number, a scalar. But in this section of the course we're going to take a look at the cousin of the dot product, the vector product, or also known as the cross product. It also takes as input two vectors, but it returns a third vector and so the utility of the cross product is that it gives us a special kind of vector that results from the input of two other vectors. We will not get a number, we will get something that has both direction and magnitude from this product. Now the cross product or vector product is a bit more complicated than its cousin, the dot product. Like the dot product, it involves multiplying two vectors, which for the purposes of this portion of the lecture I'll label as v1 vector and v2 vector. Now unlike the dot product, the result of this new multiplicative activity is a vector, not a number. We represent this special product as follows. Vector v1 and vector v2 are cross multiplied and the symbol for cross multiplication is the classic arithmetic times symbol, an x or a cross. Now the cross is crucially important here because when you write v1 cross sign and then v2, you mean very explicitly the vector product or the cross product. The dot product or scalar product is independently represented by placing a dot symbol between these two. It is really important to keep your notation straight here and also to know what notation you are seeing when you read about the product of two vectors because you need to be able to distinguish when you should be doing the scalar product from when you should be doing the vector product of two vectors. This symbol is the symbol for cross product and you should accept absolutely no substitutions. Now I've drawn over here in a three-dimensional coordinate system with an x-axis, a y-axis and a z-axis, these two vectors v1 vector shown in blue and v2 vector shown in red. Now it's perhaps a little bit difficult to see as it almost always is in two-dimensional representations of three-dimensional spaces but v1 and v2 have components that lie only and entirely in the plane defined by the x and y-axes. They have no third component in z and I emphasize that now because when we do the cross product you'll see that something very interesting happens with these two vectors. Now before we proceed, it's important to know the rules involving the cross product of unit vectors that lie along the coordinate axes. So for instance, i hat lies along the x direction and only along the x direction, j hat lies along the y direction and only along the y direction, k hat lies only along the z direction. These three unit vectors, i hat, j hat and k hat stand in for vectors that point only and entirely along one coordinate axes and they are all at right angles to each other. i hat is perpendicular to j hat, j hat is perpendicular to k hat and i hat is perpendicular to k hat. So they are very special vectors and that's why it's important to learn the rules of the cross product for these vectors first. They're the building blocks of all other vectors. So these rules for i hat, j hat and k hat are as follows. The cross product of i hat and j hat in that order, i hat cross j hat gives you k hat. In other words, the cross product of a vector that lies only along the x direction with one that lies only along the y direction gives you a vector that points only along the z direction and you can already see the utility of the cross product. If you have two vectors like i hat and j hat and you need a third vector that is perpendicular to those two, the cross product will help you. It's a very nice mathematical operation. So i hat cross j hat equals k hat. Now how do we get the rules in general for the unit vectors? Well, the trick to getting the rules for all of these unit vectors and how they multiply each other is to sort of think of i, j and k as laying on a conveyor belt, a belt where when one symbol falls off the right hand side of the machinery, it comes back and becomes part of the left hand side of the machinery and the other symbol shift forward in this system. So for instance, I can conveyor belt i, j and k so that k is now on the left, i takes the place of j in the middle and j takes the place of k on the right. So again, compare the first line to the second line. What's happened? The symbol k has fallen off the end of this equation and reappeared at the beginning. i has shifted over one position and j has shifted over one position. The symbols remain in the same places. The cross product symbol and the equal sign have not moved. So you can think of this equation as a machine where the pieces that fall off the right hand side reappear on the left and all the other pieces shift to the right by one space. This is known as a permutation. A permutation of these symbols yields the second equation. k hat cross i hat is j and then we can permute them or conveyor belt them one more time. We put j at the beginning, we shift k to the right, we shift i to the right and we get j cross k equals i. Now I'm dropping the hats here to speed this up a little bit but you get the rhythm of this. i cross j is k, k cross i is j, j cross k is i. Now if you conveyor this one more time, what equation do you get? Well if you noted that if i falls off the right hand side and then reappears on the left, j shifts over one, k shifts over one, that you get back to i cross j is k, then you're correct. In fact, this cyclic permutation of these three symbols will yield automatically from the first equation all of the rules that I've shown you so far. But these aren't the only rules you can imagine. For instance, if i hat cross j hat is k hat, well then what is j hat cross i hat? Notice that j hat cross i hat appears nowhere in these three equations. What is that cross product equal to? Well the answer lies in the following. If you swap any of the two symbols on the left hand sides of the previous equations, so i cross j, k cross i, j cross k, and change them to j cross i, i cross k, and k cross j, then you will get a negative sign appearing in front of the symbol on the right hand side of the equation. So if i cross j equals k, j cross i is negative k, and so forth. So again, if you can remember that i cross j equals k, and that simply conveyor belting or cyclically permuting those symbols in their positions in the equation will yield you three of the equations, the next three can be obtained by taking the first three, swapping the order of the multiplication on the left, and a minus sign appears on the right hand side. In other words, if you have j cross i, you get a negative k hat that points in the negative direction along the z-axis, not the positive direction that i cross j yielded, and this makes some intuitive sense. If I swap the order in which I multiply these two vectors, and I'm getting a vector out of them, it makes sense that if I swap the order of the vectors, I flip the resulting vector, right? There's some sort of intuition there that you can imagine geometrically. But we're not done yet. We have the rules for i cross j, k cross i, and j cross k, and then from this we derive the rules for j cross i, i cross k, and k cross j, as shown on this slide. But what about i cross i? What about j cross j? What about k cross k? They haven't appeared anywhere in here so far. These are maybe the easiest to remember. The cross product of any vector with itself is zero. i cross i, j cross j, k cross k, all zero. And you'll begin to see why this isn't a bit, but this is the rule for this multiplication, and this is not so bad, it's easy to commit to memory. Now notice that the cross product rules are slightly different from the dot product rules. In the dot product, i dot i yielded the number one. j dot j yielded the number one, and so forth. Here i cross i gives you zero, j cross j gives you zero. When you have the cross product of a vector and itself, it completely goes away. When you have the dot product of a vector and itself, you basically get back its length squared. So this is a little bit different than the dot product. And you can almost imagine that sort of the rules that apply to the dot product are the sort of reverse of the rules that apply to the cross product. And that might help you to remember this stuff as well, but practice is the best way to build memory. Now with those rules in mind, let's actually do the cross product of v1 and v2 shown over here on the left side of the slide. Now this involves the product of the two vectors, and we're going to distribute the cross product multiplication terms, and then keep using those aforementioned rules to simplify. Now I'm going to keep this in algebraic notation. I actually haven't given you the numbers that represent these vectors, but I'll show you how I drew them in a moment. I'm going to keep this algebraic. I'm going to substitute numbers in at the very end, so we can actually get a number for this vector, and then I can draw it for you. So the cross product is v1 cross v2. If I substitute in with the x and y components of v1 and the x and y components of v2, remember I told you, neither of these two vectors has a component in the third dimension, z. Then I would get an equation that looks like this. vx1 i hat plus vy1 j hat is how one would represent in unit vector notation the first vector v1. It's got an x component along the i hat direction. It's got a y component along the j hat direction. No real surprises here. This is going to be then cross, multiplied cross producted with the components of the second vector v2. So here we have vx2 i hat plus vy2 j hat. Okay, well, again, these are the sum of two terms, multiplied by the sum of two other terms. This is just polynomial multiplication. We can bust out our algebra at this point and start multiplying components through this to get a four term polynomial. It's not pretty. We get vx1 i hat cross vx2 i hat. That's the product of these two terms. We get vx1 i hat cross vy2 j hat. So that's the multiplication of this term with the second term and the second vector. And so forth. The remaining two terms just come from repeating this exercise with the vy1 j hat distribution into the second vector. So again, we get a four term polynomial. Each term is the cross product of two components of these two vectors. And so we wind up with four total terms. Now, I can play the old game that scalar numbers like vx1 and vx2, they don't play a role in the vector product, either dot or cross. So I can pull them out in front and then isolate the cross product itself. And that's what I'm going to do in the next line of this equation. I pulled, for instance, vx1 times vx2 out in front. And this leaves me with i hat cross i hat here in parentheses. And repeat that for the second term, vx1 vy2. And I get i hat cross j hat, vy1 vx2, and I get j hat cross i hat. Again, order here is important. You must preserve the order in your cross product. Don't start swapping things around in the cross product without thinking. Or you'll get yourself into a lot of trouble very quickly. Keep the order in the cross product the same as the way you wrote it in the earlier line. And then finally, we have vy1 vy2 and j hat cross j hat. Now, this is where we need to bust out our rules for the cross products of unit vectors in the coordinate system. What is the cross product of i hat with itself? Well, the cross product of any unit vector with itself is always zero. So this term will vanish. And also the last term will vanish, j hat cross j hat is zero. I can indicate that here in the next line of this equation. Now we've eliminated two of the four terms. That's nice, but we're left with two terms. So let's see what we can do to simplify and group these. Well, I notice that this term in the equation involves i hat cross j hat. Think back to the rules, i hat cross j hat equals k hat. So this can be replaced with k hat, a vector that points only and entirely in the positive z direction. I have your j hat cross i hat. Well, that's this, but reversed. What happens when you reverse the order of cross multiplication? Well, you get that third vector, but with a sign flip. So we don't get k hat here, we get negative k hat. So we wind up with an equation that looks like this. Vx1 vy2 k hat plus vy1 vx2 negative k hat. Well, I've got k hat and I've got k hat. I can group these terms one more time, pulling the k hat out in front of everything, and I will get this final, and pretty much as far as I can take it simplified equation, which is a number out in front of a unit vector. The number is vx1 vy2 minus vy1 vx2. That's the number I get, where each of these is the number that represents the length of the component of that piece of the vector. And then finally, I just have k hat, which indicates the direction of this cross product vector. Now, again, notice something remarkable here. We started off with vectors that only had components in the x and y direction, and we end with a vector that only has a component in the z direction. This hints at a key property of the cross product, which I hinted at earlier. It returns a vector that is exactly perpendicular to the other two. I'll come back to that and re-emphasize that in a moment. But that is one of the key aspects of the cross product. If you have two vectors and you need a third vector perpendicular to those two, the cross product is your friend. So indeed, we've obtained a vector, as promised from the vector product or cross product in the first place. Now, what vector do we actually get? Well, for the drawing at the left, I actually used some specific numbers to draw v1 and v2. v1 is 1 comma 2. It has one unit of length along the x direction, two units of length along the y direction. And you can see that reflected here in the blue vector. v2 was 2 comma 1. It's sort of the reverse. It's got two units of length along the x direction, one unit of length along the y direction. And if we insert these numbers into our cross product, which I'll rewrite here from the previous slide, the magnitude of this cross product out in front, vx1, vy2, minus vy1, vx2, plug in the numbers there. We get one times one minus two times two, which yields negative three. So numerically, the vector that I drew here over on the left, the cross product of v1 and v2 is negative three k hat. And so if I draw that here, this is the resulting vector capital V from the cross product. It points in the negative z direction. It has a length of three, and I've represented it here in this three-dimensional coordinate system. So again, I wanna emphasize that the angle between big V and v2 is 90 degrees. And the angle between big V and v1 is 90 degrees. Big V is exactly perpendicular to either of v1 or v2. So we have obtained a vector that has no components in x or y. It lies purely along the negative z axis, and it has a length of three. That's a lot of algebra gymnastics. Is there a quicker way we could at least estimate, for instance, the magnitude or the direction of the cross product? Well indeed, let's take a closer look at some of the properties of the cross product, and these answers will come up. So what the heck is the vector represented by the cross product? It's great that we got one, but what exactly are its properties and what sort of physically does it mean? Well, the cross product always returns a vector, and that vector is always perpendicular to the two vectors used in the product. So go ahead and check this for yourself. Take the vector I just got, big V, negative 3K hat, dot product it with either v1 or v2. If you have two vectors with a 90 degree angle between them, the dot product always yields zero. Remember the magnitude of the length of a dot product goes as the cosine of the angle between them. The cosine of 90 degrees is zero. So if indeed there's a 90 degree angle between big V vector and either v1 vector or v2 vector, the dot product should yield zero in both of those cases. Go ahead and check it, and you'll see that it does. And that's because the original two vectors v1 and v2 involved only i hat and j hat. The third vector, its cross product involves only k hat, and k hat dotted into i hat is zero. You'll see that it's exactly true. But in general, you'll find that this is always true of any vector, big V, that results from the cross product of two little vectors, v1 and v2. The dot products are exactly zero. Now what about its length? Well, its length in general is given by the following equation. The length of the cross product, that's what this symbol here on the left represents, is equal to the length of v1 times the length of v2 times the sine of theta, times the sine of theta, where theta is the angle between v1 and v2. That's just as in the dot product, it's the angle between v1 and v2 that enters into this calculation. Now in the dot product, v1 dot v2, the resulting number represented the physically the following. It was the length of one vector, say v1, multiplied by the component of the second vector, v2 cosine theta, that lies along the first. That's what v2 cosine theta was. It's a projection of v2 onto v1. That's what the dot product physically represented. The vector product represents something slightly different. It returns, first of all, a vector. What's the length of the vector? Well, the length of the vector is given by the length of one vector, say v1, multiplied by the component of the second vector, v2, that is perpendicular to the first one. v2 sine theta is the length of the piece of v2 that points at a right angle to v1. Do some trigonometry and play along with this a little bit and you'll see how this feels a bit more. We'll practice this, of course, as we go forward in the course. This will come immensely in handy for rotational motion, deeper aspects of rotational motion, especially rotational forces and rotational momentum. Now, is it possible to determine the direction of the cross product without doing all of those algebraic gymnastics? The answer is yes. There is a helpful rule that you can use to aid in finding the direction of the cross product and it is another kind of right-hand rule. Now, when we looked at rotation earlier in a more general sense beyond uniform circular motion, we encountered quantities like angular velocity and angular acceleration and they indeed can be represented by a vector, but that vector is perpendicular to the plane of rotation of the object. So, the cross product has a right-hand rule in the same sense that finding angular velocity or angular acceleration has an associated right-hand rule. So, let's go ahead and talk about this. It's depicted here over on the left and I'll walk you through the stages of it in a moment. But let's imagine that we have two vectors, A and B, and we want to take the cross product, specifically in the order of A cross B, A first, then B. We want to get C and we want to know the direction that C points in. So, take your right-hand, take the index finger on your right-hand, also known as the pointer finger, so it's this first finger next to the thumb, pointed in the direction of the first vector in the product. Well, the first vector in the product is A vector here. So, take the index finger, the pointer finger on your right-hand, the one closest to your thumb, and point it right in the direction of A. Now, you're going to take the middle finger on your right-hand as depicted here. That's the next finger on your right-hand. And you're going to try to point it in the direction of B while maintaining your index finger pointing along A. Don't start rotating your hand randomly. You do need to reorient your hand, potentially, but your goal is to keep your index finger pointed in the direction of A and orient your middle finger so that it indicates the direction of B, the second vector in the cross product. To do this, you're going to have to tilt your hand around and we'll do an exercise in this in a moment because your goal is to comfortably point your index finger along A and your middle finger along B. Remember, your fingers really only bend in one direction and so please do not try to crack your knuckles to make your fingers bend in directions they were not intended to go. Just tilt your hand, rotate your wrist so that your middle finger points in such a way that it comfortably indicates the direction of B while maintaining the index finger pointing in the direction of A. Now, stick your thumb out, perpendicular to both your index and middle finger and that's indicated in the picture here. The thumb now indicates the direction that the cross product C vector points in. You now have C hat, the unit vector that indicates the direction where C or A cross B points. That's the right hand rule for the cross product. Point your index finger in the direction of the first vector while maintaining that pointing get your middle finger to point in the direction of the second vector. You may have to rotate your hand, flip your hand over so that you can comfortably do this, that's okay. Flipping is okay. And then stick your thumb out perpendicular to both of those fingers and the result is that your thumb points in the direction of the cross product. It's not too bad but you should practice it and we're gonna practice it right now. So, to practice the right hand rule for cross product multiplication I want you to try to answer the following question and after I ask the question I want you to pause the video and play around with this keeping the rules of the right hand rule for cross products in mind. So, the question is the following given the vectors A and B shown below A here in blue, B here in red what is the direction of the cross product A cross B is it into or out of the screen? So, keep the rules of the right hand rule for cross product multiplication in mind. You're gonna take your index finger you're gonna point it in the direction of A now you may have to flip your hand over from where it's currently oriented but your goal is to now get your middle finger to point in the direction of B stick your thumb out perpendicular to the two your thumb indicates the direction of the cross product A cross B or C vector if you wanna call it that. Now, once you've done that I want you to answer the question does this resulting vector of the cross product point into the screen or out of the screen? So, pause the video here and try this out for yourself and resume the video when you're ready to see the answer. The answer is that you should be forced to flip your hand over so that your thumb points down into the screen and I'm indicating that direction here with a circle with a cross through it. Vectors are like arrows and if you think about an arrow in the sense of an archery arrow where you have a long stick with a point on one end and some kind of stabilizing feathers on the backside you draw the bow back and you fire this arrow it's difficult to represent three dimensions very easily in a two dimensional surface like the screen and so the way that we do this notationally is if we wanna indicate a direction in the third dimension into or out of the screen in this case we use a circle with an arrow in it for a vector that points into the screen and we use a circle with a dot in it to indicate a vector that points out of the screen and this is analogous to the stabilizing tail feathers of an archery arrow flying away from you so into the screen the tail feathers would make a little cross and that's what the cross is kind of meant to represent but if the arrow is coming out of the screen at you you see the point of the arrow and that would be the circle with the dot in the middle of it so that's just a mathematical graphical notational trick let me write that down here if I indicate a circle with a cross through it like I've drawn here it means a vector pointing into the screen here's that symbol written out for you separately from the coordinate system here if instead I had drawn a circle with a dot in it it means a vector pointing out of the screen and that's indicated here now for this problem again in order to answer the question where does the cross product point you have to point your index finger in the direction of A you're forced to flip your hand over to get your middle finger to point roughly in the direction of B your thumb then points down into the screen and so to represent that direction down into the screen I use this symbol a circle with a cross through it now as a bonus question you could answer the following what direction does the cross product B cross A point in so you could try that as a variation on the above question so A and B still are as drawn but now calculate the direction of B cross A using the right hand rule you should find that now your thumb points out of the screen toward you so B cross A points in the opposite direction of A cross B and again that makes sense whenever you flip the order in which you do the cross product you flip the result in cross product vector just as we saw with the unit vector rules earlier in this video so let's review the key ideas that we have learned in this section of the course vectors as we have now fully emphasized have two kinds of multiplication we've reminded ourselves that the scalar product or dot product takes as input to vectors but it returns a number just a scalar number could be positive or negative but it's just a number however we now have the second kind of multiplication for vectors the vector product or cross product and this can be used to construct a third vector from those two original vectors and we've seen some of the properties of this we've learned the formula for calculating the length of the result in cross product so if we have two vectors A vector and B vector the length is AB sine theta where theta is the angle between A and B and we've learned a right hand rule for this in addition to all the algebraic rules for the actual unit vectors to help us find the direction of the third vector very quickly the right hand rule can be used to very quickly check a calculation where you have ground through the algebra of I cross J, J cross I, I cross K and so forth it's good to help build some intuition to use the right hand rule in conjunction with a full algebraic calculation so that you have a comfort zone by calculating these things we're going to use this cross product going forward it's a necessary ingredient in understanding rotational motion and more to the point describing the vectors associated with rotational momentum and also forces associated with rotations