Dot vs. cross product

Let's do a little compare and contrast between the dot product and the cross product. Let me just make two vectors-- just visually draw them. And maybe if we have time, we'll, actually figure out some dot and cross products with real vectors. Let's call the first one-- That's the angle between them. OK. So let's just go over the definitions and then we'll work on the intuition. And hopefully, you have a little bit of both already. So what is a dot b? Well first of all, that's the exact same thing as b dot a. Order does not matter when you take the dot product because you end up with just a number. And that is equal to the magnitude of a times the magnitude of b times cosine of the angle between them. Let's look at the definition of the cross product. What is a cross b? Well first of all, that does not equal b cross a. It actually equals the opposite direction, or you could view it as the negative of b cross a. Because the vector that you end up with ends up flipped, whichever order you do it in. But a cross b, that is equal to the magnitude of vector a times the magnitude of vector b-- so far, it looks a lot like the dot product, but this is where the diverge is-- times the sine of the angle between them. The sine of the angle between them. And this is where it really diverges. When we took the dot product, we just ended up with a number. This is just a number. There's no direction here. This is just a scalar quantity. But the cross product, we take the magnitude of a times the magnitude of b, times the sine of the angle between them, and that provides a magnitude, but it also has a direction. And that direction is provided by this normal vector. It's a unit vector. A unit vector gets that little hat on it. It's a unit vector, and what direction is it? Well, that's defined by the right hand rule. This is a vector. It's perpendicular to both a and b. And then you might say, a and b, the way I drew them, they're both sitting in the plane of this video screen, or your video screen. So in order for something to be perpendicular to both of these, it either has to pop out of the screen or pop into the screen, right? And when you learned about the cross product, I said there's two ways of showing a vector popping out of the screen. It looks like that because that's the tip of an arrow. And to show a vector going into the screen, it's like that because that is the back of an arrow. The rear end of an arrow. So how do you know which of these two it is? Because both of these vectors are perpendicular to a and b. That's where you take your right hand and you use the right hand rule. So you take your index finger in the direction of a, your middle finger in the direction of b, and then your thumb points in the direction of n. So let's do that. I'm looking at my hand. It's not an easy thing to do with your right hand, but your right hand is going to look something like this. Your index finger will go in the direction of a. Your middle finger goes in the direction of b. So that's my middle finger. And then my other two fingers just do what they need to do. I like to just bend them out of the way. So they just curl around my hand. And then what direction is my thumb in? My thumb-- well, actually I drew it at the wrong angle. My thumb is actually going in this direction, right? Into the page. This is the top of my hand. These are like my veins. Or, if I actually drew it correctly, where you would see your hand from side-- so it would look like this. You would see your pinky. Your palm and your pinky would be like that. And your other finger like this. Your middle finger would go in the direction of b. Your index finger goes in the direction of a, and you wouldn't even see your thumb, because your thumb is pointing straight down. But I think you get the point. a cross b, this n vector is pointing straight down. It's a unit vector. And this provides the magnitude. Unit vector just means it has a magnitude of one. So the magnitudes of the cross and the dot products seem pretty close. They both have the magnitude of both vectors there. Dot product, cosine theta. Cross product sine of theta. But then, the huge difference is that sine of theta has a direction. It is a different vector that is perpendicular to both of these. Now, let's get the intuition. And if you've watched the videos on the dot and the cross product, hopefully you have a little intuition. But I review it because I think it all fits together when you see them with each other. First, let's study a, b cosine of theta. If you watched the dot product video, cosine of theta, if you took, let's say, b cosine of theta. What is b cosine of theta? b cosine of theta-- and you could work it out on your own time-- if you say cosine is adjacent over hypotenuse, the magnitude of b cosine theta is actually going to be the magnitude of, if you dropped a perpendicular-- I'll use a different color here-- if you dropped a perpendicular here, this length right here-- that's b cosine theta. Let me draw it separately. I don't want to mess up this picture too much. So if that's b. If that's a-- And that's b. That's a. This is theta. b cosine theta, if you drop a line perpendicular to a, this is a right angle. b cosine theta, adjacent over hypotenuse is equal to cosine theta. So it would be the projection of b going in the same direction as a. So it would be this magnitude. That is b cosine theta. So the magnitude of that vector right there is the magnitude of b cosine of theta. So when you're taking the dot product, at least the example I just did, if you view it as the magnitude of a times the magnitude of b cosine theta, you're saying what part of b goes in the same direction as a? And whatever that magnitude is, let me just multiply that times the magnitude of a. And I have the dot product. Let's take the pieces that go the same direction and multiply them. So how much do they move together? Or do they point together? Or you could view it the other way. You could view the dot product as-- and I did this in the dot product video-- you could view it as a cosine of theta, b. Because it doesn't matter. These are all scalar quantities, so it doesn't matter what order you take the multiplication in. And a cosine theta is the same thing. It's the magnitude of the a vector that's going in the same direction of b. Or the projection of a onto b. So this vector right here is a cosine theta; the magnitude of a cosine theta. And they're actually the same number. If you take how much of b goes in the direction of a, and multiply that with the magnitude of a, that gives you the same number as how much of a goes in the direction of b, and then multiply the two magnitudes. Now, what is a, b sine theta? a, b, sine theta. Well if this vector right here is a cosine theta-- and you learned this when you learned how to take the components of vectors. This vector right here is the magnitude of a sine theta. You could rewrite this as the magnitude of a sine theta times the magnitude of b in that normal vector direction. So if you take a sine theta times b, you're saying what part of a doesn't go the same direction as b. What part of a is completely perpendicular to b-- has nothing to do is b. They share nothing in common. It goes in a completely different direction. That's a sine theta. And so you take the product of this with b and then you get a third vector. And it almost says, how different are these two vectors? And it points in a different direction. It gives you this-- sometimes it's called a pseudo vector, because it applies to some concepts that are pseudo vectors. But the most important of these concepts is torque, when we talk about the magnetic field; the force of a magnetic field on electric charge. These are all forces, or these are all physical phenomena, where what matters isn't the direction of the force with another vector, it's the direction of the force perpendicular to another vector. And so that's where the cross product comes in useful. Anyway, hopefully, that gave you a little intuition. And you could have done it the other way. You could have written this as b sine theta. And then you would have said that's the component of b that is perpendicular to a. So b sine theta actually would have been this vector. Or let me draw it here. That would make more sense. This would be b sine theta. So you could switch orders. You could visualize it either way. You could say this is the magnitude of b that is completely perpendicular to a, multiply the two, and use the right hand rule to get that normal vector. And we just decided that we're going to use the right hand rule to have a common convention. But people could have used the left hand rule, or they might have used it a different way. It's just a way that we have a consistent framework, so that when we take the cross product we all know what direction that normal vector is pointing in. Anyway. In the next video I'll show you how to actually compute dot and cross products when you're given them in their component notation. See you in the next video.