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Cross product 1

Introduction to the cross product. Created by Sal Khan.

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Video transcript

I've been requested to do a video on the cross product, and its special circumstances, because I was at the point on the physics playlist where I had to teach magnetism anyway, so this is as good a time as any to introduce the notion of the cross product. So what's the cross product? Well, we know about vector addition, vector subtraction, but what happens when you multiply vectors? And there's actually two ways to do it: with the dot product or the cross product. And just keep in mind these are-- well, really, every operation we've learned is defined by human beings for some other purpose, and there's nothing different about the cross product. I take the time to say that here because the cross product, at least when I first learned it, seemed a little bit unnatural. Anyway, enough talk. Let me show you what it is. So the cross product of two vectors: Let's say I have vector a cross vector b, and the notation is literally like the times sign that you knew before you started taking algebra and using dots and parentheses, so it's literally just an x. So the cross product of vectors a and b is equal to-- and this is going to seem very bizarre at first, but hopefully, we can get a little bit of a visual feel of what this means. It equals the magnitude of vector a times the magnitude of vector b times the sine of the angle between them, the smallest angle between them. And now, this is the kicker, and this quantity is not going to be just a scalar quantity. It's not just going to have magnitude. It actually has direction, and that direction we specify by the vector n, the unit vector n. We could put a little cap on it to show that it's a unit vector. There are a couple of things that are special about this direction that's specified by n. One, n is perpendicular to both of these vectors. It is orthogonal to both of these vectors, so we'll think about it in a second what that implies about it just visually. And then the other thing is the direction of this vector is defined by the right hand rule, and we'll see that in a second. So let's try to think about this visually. And I have to give you an important caveat: You can only take a cross product when we are dealing in three dimensions. A cross product really has-- maybe you could define a use for it in other dimensions or a way to take a cross product in other dimensions, but it really only has a use in three dimensions, and that's useful, because we live in a three-dimensional world. So let's see. Let's take some cross products. I think when you see it visually, it will make a little bit more sense, especially once you get used to the right hand rule. So let's say that that's vector b. I don't have to draw a straight line, but it doesn't hurt to. I don't have to draw it neatly. OK, here we go. Let's say that that is vector a, and we want to take the cross product of them. This is vector a. This is b. I'll probably just switch to one color because it's hard to keep switching between them. And then the angle between them is theta. Now, let's say the length of a is-- I don't know, let's say magnitude of a is equal to 5, and let's say that the magnitude of b is equal to 10. It looks about double that. I'm just making up the numbers on the fly. So what's the cross product? Well, the magnitude part is easy. Let's say this angle is equal to 30 degrees. 30 degrees, or if we wanted to write it in radians, I always-- just because we grow up in a world of degrees, I always find it easier to visualize degrees, but we could think about it in terms of radians as well. 30 degrees is-- let's see, there's 3, 6-- it's pi over 6, so we could also write pi over 6 radians. But anyway, this is a 30-degree angle, so what will be a cross b? a cross b is going to equal the magnitude of a for the length of this vector, so it's going to be equal to 5 times the length of this b vector, so times 10, times the sine of the angle between them. And, of course, you could've taken the larger, the obtuse angle. You could have said this was the angle between them, but I said earlier that it was the smaller, the acute, angle between them up to 90 degrees. This is going to be sine of 30 degrees times this vector n. And it's a unit vector, so I'll go over what direction it's actually pointing in a second. Let's just figure out its magnitude. So this is equal to 50, and what's sine of 30 degrees? Sine of 30 degrees is 1/2. You could type it in your calculator if you're not sure. So it's 5 times 10 times 1/2 times the unit vector, so that equals 25 times the unit vector. Now, this is where it gets, depending on your point of view, either interesting or confusing. So what direction is this unit vector pointing in? So what I said earlier is, it's perpendicular to both of these. So how can something be perpendicular to both of these? It seems like I can't draw one. Well, that's because right here, where I drew a and b, I'm operating in two dimensions. But if I have a third dimension, if I could go in or out of my writing pad or, from your point of view, your screen, then I have a vector that is perpendicular to both. So imagine of vector that's-- I wish I could draw it-- that is literally going straight in at this point or straight out at this point. Hopefully, you're seeing it. Let me show you the notation for that. So if I draw a vector like this, if I draw a circle with an x in it like that, that is a vector that's going into the page or into the screen. And if I draw this, that is a vector that's popping out of the screen. And where does that convention come from? It's from an arrowhead, because what does an arrow look like? An arrow, which is our convention for drawing vectors, looks something like this: The tip of an arrow is circular and it comes to a point, so that's the tip, if you look at it head-on, if it was popping out of the video. And what does the tail of an arrow look like? It has fins, right? There would be one fin here and there'd be another fin right there. And so if you took this arrow and you were to go into the page and just see the back of the arrow or the behind of the arrow, it would look like that. So this is a vector that's going into the page and this is a vector that's going out of the page. So we know that n is perpendicular to both a and b, and so the only way you can get a vector that's perpendicular to both of these, it essentially has to be perpendicular, or normal, or orthogonal to the plane that's your computer screen. But how do we know if it's going into the screen or how do we know if it's coming out of the screen, this vector n? And this is where the right hand rule-- I know this is a little bit overwhelming. We'll do a bunch of example problems. But the right hand rule, what you do is you take your right hand-- that's why it's called the right hand rule-- and you take your index finger and you point it in the direction of the first vector in your cross product, and order matters. So let's do that. So you have to take your finger and put it in the direction of the first arrow, which is a, and then you have to take your middle finger and point it in that direction of the second arrow, b. So in this case, your hand would look something like this. I'm going to try to draw it. This is pushing the abilities of my art skills. So that's my right hand. My thumb is going to be coming down, right? That is my right hand that I drew. This is my index finger, and I'm pointing it in the direction of a. Maybe it goes a little bit more in this direction, right? Then I put my middle finger, and I kind of make an L with it, or you could kind of say it almost looks like you're shooting a gun. And I point that in the direction of b, and then whichever direction that your thumb faces in, so in this case, your thumb is going into the page, right? Your thumb would be going down if you took your right hand into this configuration. So that tells us that the vector n points into the page. So the vector n has magnitude 25, and it points into the page, so we could draw it like that with an x. If I were to attempt to draw it in three dimensions, it would look something like this. Vector a. Let me see if I can give some perspective. If this was straight down, if that's vector n, then a could look something like that. Let me draw it in the same color as a. a could look something like that, and then b would look something like that. I'm trying to draw a three-dimensional figure on two dimensions, so it might look a little different, but I think you get the point. Here I drew a and b on the plane. Here I have perspective where I was able to draw n going down. But this is the definition of a cross product. Now, I'm going to leave it there, just because for some reason, YouTube hasn't been letting me go over the limit as much, and I will do another video where I do several problems, and actually, in the process, I'm going to explain a little bit about magnetism. And we'll take the cross product of several things, and hopefully, you'll get a little bit better intuition. See you soon.