A little more intuition on the cross product. Created by Sal Khan.
Let's see if we can get a little bit more practice and intuition of what cross products are all about. So in the last example, we took a cross b. Let's see what happens when we take b cross a. So let me erase some of this. I don't want to erase all of it because it might be useful to give us some intuition to compare. I'm going to keep that. Actually, I can erase this, I think. So the things I have drawn here, this was a cross b. Let me cordon it off so you don't get confused. So that was me using the right hand rule when I tried to do a cross b, and then we saw that the magnitude of this was 25, and n, the direction, pointed downwards. Or when I drew it here, it would point into the page. So let's see what happens with b cross a, so I'm just switching the order. b cross a. Well, the magnitude is going to be the same thing, right? Because I'm still going to take the magnitude of b times the magnitude of a times the sine of the angle between them, which was pi over 6 radians and then times some unit vector n. But this is going to be the same. When I multiply scalar quantities, it doesn't matter what order I multiply them in, right? So this is still going to be 25, whatever my units might have been, times some vector n. And we still know that that vector n has to be perpendicular to both a and b, and now we have to figure out, well, is it, in being perpendicular, it can either kind of point into the page here or it could pop out of the page, or point out of the page. So which one is it? And then we take our right hand out, and we try it again. So what we do is we take our right hand. I'm actually using my right hand right now, although you can't see it, just to make sure I draw the right thing. So in this example, if I take my right hand, I take the index finger in the direction of b. I take my middle finger in the direction of a, so my middle figure is going to look something like that, right? And then I have two leftover fingers there. Then the thumb goes in the direction of the cross product, right? Because your thumb has a right angle right there. That's the right angle of your thumb. So in this example, that's the direction of a, this is the direction of b, and we're doing b cross a. That's why b gets your index finger. The index finger gets the first term, your middle finger gets the second term, and the thumb gets the direction of the cross product. So in this example, the direction of the cross product is upwards. Or when we're drawing it in two dimensions right here, the cross product would actually pop out of the page for b cross a. So I'll draw it over. It would be the circle with the dot. Or if I were to draw it analogous to this, so this right here, that was a cross b. And then b cross a is the exact same magnitude, but it goes in the other direction. That's b cross a. It just flips in the opposite direction. And that's why you have to use your right hand, because you might know that, oh, something's going to pop in or out of the page, et cetera, et cetera, but you need to know your right hand to know whether it goes in or out of the page. Anyway, let's see if we can get a little bit more intuition of what this is all about because this is all about intuition. And frankly, I'll tell you, the cross product comes into use in a lot of concepts that frankly we don't have a lot of real-life intuition, with electrons flying through a magnetic field or magnetic fields through a coil. A lot of things in our everyday life experience, maybe if we were metal filings living in a magnetic field-- well, we do live in a magnetic field. In a strong magnetic field, maybe we would get an intuition, but it's hard to have as deep of an intuition as we do for, say, falling objects, or friction, or forces, or fluid dynamics even, because we've all played with water. But anyway, let's get a little bit more intuition. And let's think about why is there that sine of theta? Why not just multiply the magnitudes times each other and use the right hand rule and figure out a direction? What is that sine of theta all about? I think I need to clear this up a little bit just so this could be useful. So why is that sine of theta there? Let me redraw some vectors. I'll draw them a little fatter. So let's say that's a, that's a, this is b. b doesn't always have to be longer than a. So this is a and this is b. Now, we can think of it a little bit. We could say, well, this is the same thing as a sine theta times b, or we could say this is b sine theta times a. I hope I'm not confusing-- all I'm saying is you could interpret this as-- because these are just magnitudes, right? So it doesn't matter what order you multiply them in. You could say this is a sine theta times the magnitude of b, all of that in the direction of the normal vector, or you could put the sine theta the other way. But let's think about what this would mean. a sine theta, if this is theta. What is a sine theta? Sine is opposite over hypotenuse, right? So opposite over hypotenuse. So this would be the magnitude of a. Let me draw something. Let me draw a line here and make it a real line. Let me draw a line there, so I have a right angle. So what's a sine theta? This is the opposite side. So a sine theta is a, and sine of theta is opposite over hypotenuse. The hypotenuse is the magnitude of a, right? So sine of theta is equal to this side, which I call o for opposite, over the magnitude of a. So it's opposite over the magnitude of a. So this term a sine theta is actually just the magnitude of this line right here. Another way you could-- let me redraw it. It doesn't matter where the vectors start from. All you care about is this magnitude and direction, so you could shift vectors around. So this vector right here, and you could call it this opposite vector, that's the same thing as this vector. That's the same thing as this. I just shifted it away. And so another way to think about it is, it is the component of vector a, right? We're used to taking a vector and splitting it up into x- and y-components, but now we're taking a vector a, and we're splitting it up into-- you can think of it as a component that's parallel to vector b and a component that is perpendicular to vector b. So a sine theta is the magnitude of the component of vector a that is perpendicular to b. So when you're taking the cross product of two numbers, you're saying, well, I don't care about the entire magnitude of vector a in this example, I care about the magnitude of vector a that is perpendicular to vector b, and those are the two numbers that I want to multiply and then give it that direction as specified by the right hand rule. And I'll show you some applications. This is especially important-- well, we'll use it in torque and we'll also use it in magnetic fields, but it's important in both of those applications to figure out the components of the vector that are perpendicular to either a force or a radius in question. So that's why this cross product has the sine theta because we're taking-- so in this, if you view it as magnitude of a sine theta times b, this is kind of saying this is the magnitude of the component of a perpendicular to b, or you could interpret it the other way. You could interpret it as a times b sine theta, right? Put a parentheses here. And then you could view it the other way. You could say, well, b sine theta is the component of b that is perpendicular to a. Let me draw that, just to hit the point home. So that's my a, that's my b. This is a, this is b. So b has some component of it that is perpendicular to a, and that is going to look something like-- well, I've run out of space. Let me draw it here. If that's a, that's b, the component of b that is perpendicular to a is going to look like this. It's going to be perpendicular to a, and it's going to go that far, right? And then you could go back to SOH CAH TOA and you could prove to yourself that the magnitude of this vector is b sine theta. So that is where the sine theta comes from. It makes sure that we're not just multiplying the vectors. It makes sure we're multiplying the components of the vectors that are perpendicular to each other to get a third vector that is perpendicular to both of them. And then the people who invented the cross product said, well, it's still ambiguous because it doesn't tell us-- there's always two vectors that are perpendicular to these two. One goes in, one goes out. They're in opposite directions. And that's where the right hand rule comes in. They'll say, OK, well, we're just going to say a convention that you use your right hand, point it like a gun, make all your fingers perpendicular, and then you know what direction that vector points in. Anyway, hopefully, you're not confused. Now I want you to watch the next video. This is actually going to be some physics on electricity, magnetism and torque, and that's essentially the applications of the cross product, and it'll give you a little bit more intuition of how to use it. See you soon.