3. Vectors

Intro to Cross Product (Vector Product)

# Vector (Cross) Product and the Right-Hand-Rule

Patrick Ford

1027

12

Was this helpful?

Bookmarked

Hey guys. So in a previous video I covered that. There was two ways of multiplying vectors. One was called the scalar product or sometimes called the dot product. We've already covered that. And in this video, I want to show you the second way to multiply vectors, which is called the vector product or sometimes called the cross product. I'm also going to cover a really important rule for cross products that you'll need to know which is called the right hand rule. And you're gonna be using this a lot sometimes in physics. So let's go ahead and check this out here and we're gonna do a bunch of examples. Now, before we begin, we begin, I just want to briefly recap dot products. So imagine I have these two vectors here, A and B. They're offset by some angle of 60 degrees. Remember the dot product or the scalar product is just one way I can multiply this three and this four vector. But what I get out of it is just a number, right? I don't get a vector out of it. So the equation that we use for this is this a B. Cosine of theta. Right? So if we just do this here, A times B or a dot B was a B. Cosine of theta. So in other words, it was just the magnitude of a which is three magnitude of B, which is four times the cosine of the angle between them, which is just 60 degrees. What you get out of this is you just get six right? You don't get a vector, you just get the number six. And what the six really just represents here is it just represents the multiplication of parallel components of A. And B. So for example this be vector here has a component that lies in the same direction parallel to A. And if you work this out you're gonna get the B. X. Component is too. So if you multiply these two components here that are parallel, you end up just getting six out of it. Alright, so that's the scalar products. Now we're gonna do the same thing, We're gonna multiply these using the vector products and the whole idea here guys, the thing that's different about the vector product is that you get a third vector out of it. You're gonna get a new vector. That's why we call it the vector product. And this vector here is gonna be called C. Alright, so with the scalar product, you multiply them, you just get a number with the vector product. We're gonna multiply these two vectors and we're gonna get a third one out of it called C. That's what you need to know about the C vector. Is that it is perpendicular to both A and B. Alright. The equation that we're gonna use to calculate the magnitude is going to be very similar to the dot product. We're going to use a times B. Times the sine of theta. So before we use the cosine. But now we're going to use the sign. So it's the magnitude of a magnitude of B. Times the sine of the angle. And this angle here, Theta represents the smallest of the angles between A. And B. So in other words there's two ways to represent this angle. You can go all the way around, but that's not the smallest angle. So we're just gonna use the 60. Alright? So if we do the vector product of these two vectors, what you're gonna get is a third vector, which is C. And the magnitude of the C vector is just gonna be a B times sine of theta. So we're gonna do the same thing we did over here, this is gonna be three and 43 times four. Except now we're gonna do sign of 30. Now, if you I'm sorry, this is a sign of 60. And if you work this out, what you're gonna get here is 10.4. The magnitude of this vector is not gonna be six, it's going to be 10.4 because now we're using this sign. Okay, but this is a vector. Remember magnetic vectors have both magnitude and direction. Where does this 10.4 vector point. And to do that, we're gonna use something called the right hand rule. So to find the direction of the vector products, we're going to use something called the right hand rule. Now different people have different rules for doing this. If your professor has a preference, I would say stick with that. But I'm gonna show you the way I think is the easiest way to do this. And I highly suggest that you do this with me every single time, because the more you do it, the more you'll get familiar with it. All right, So here's how the right hand rule works. What you're gonna do is you're gonna take your hand and you're gonna point your fingers along the first vector always you're always gonna point along the first vector, which is in our case is a. So we're gonna point our fingers along a like this. Okay, now what you wanna do is you want to curl your fingers towards the second vector. So you're gonna curl them towards B. So in other words, I'm gonna curl my fingers up like this in the direction where B. Is and what you do that, what you're going to see is that your thumb points in the direction of C. So your thumb is going to give you the direction of that new vector. So you're gonna take your hand pointed a, curl it towards B. And now my thumb is going to be pointing out towards me. So the way that we represent this on a two dimensional graph, like a you know xy xy plane is we just draw a little circle with a dot going through this this symbol here, The circle of the dot means it's going out of the page towards you, right? So you can kind of imagine like an arrow that's coming straight towards you. All you see is just the tip. Okay, what's really important about this? Is that your curling first? You're pointing your fingers along the first and then your curling towards the second one. So let me show you, for example what happens if you do it the other way around. So in this diagram I have the same exact numbers, the same exact vectors. Except now they're flipped. Right now I have a that's pointing up like this and be that's going down. Alright, so what happens here Is that the magnitude of this is still gonna be 10.4, right? And I'm still going to have another vector. But now what happens is I'm going to point my fingers along a and in order to curl towards B, I have to flip my hand around, right, you're going to do this so you're gonna have to sort of contort your hand like this. My fingers point along a curled towards B. And your thumb is now gonna be pointing away from you. My thumb is pointing away from me towards you, right? So it's pointing away from me. And so the symbol that we use for that is a circle but now we've drawn X through it like this, that indicates that it's going out of the sort of into the page away from you. Okay, so you can kind of imagine that if you were to sort of fire an arrow and look behind it, you would see a little feathers or something going away from you. Okay, so that's basically all there is to it. So you're gonna use the magnitude, you're gonna use this equation to calculate the magnitude and then you're gonna use the right hand rule to calculate the direction. Alright, so let's go ahead and just do a bunch of examples together. Alright, so now we want to do is we wanna sort of calculate the magnitude and direction of these vector products here. Now what I want to mention off the bat is that sometimes sometimes in some books you may be given sort of a three dimensional view like this. Um And some books will even do X. And y like this and then Z. And y like this. There's actually no right way to do it. It really just comes down to preference. I'm just gonna show you both of them just in case you run into them. Okay, so here's what happens. So we've got these two vectors, I've got six and five. So what I wanna do is calculate the magnitude and direction of a cross B. So in other words, when I do this we're gonna we're gonna get is that the magnitude of C. Which is gonna be a. B. Times the sine of theta. Okay, so the magnitude of a is just six. The magnitude of B is five And the sine of the angle between them. Well, what happens here is we have an an era that goes along the X axis and one that's still kind of like in the XZ plane. So what I want you to do is kind of imagine that you had two pens on like a tabletop and they're both flat but they're just sort of pointing offset. Right? So one of the pens is sort of like facing away from you But they're basically both still flats. Okay, so the angle between them is going to be the 30°. So we're gonna have six times five times the sine of 30, which is 15. Alright, so that's the magnitude, where does it points? We're gonna have to use the right hand rule. So again take your fingers, point them along A So point them sort of flat on your tabletop like this. And I want you to do is curl them away from you towards be so curl your fingers like this away from you. And what you'll see is that your thumb is going to be pointing up. So in other words, this vector here is going to be pointing along the y axis in this problem. So we've got C. Is equal to 15 and it points along the plus y axis. Okay, so the one way, one way you can kind of see this is that if you are looking at this from the top down, right? If you are looking down from the tabletop then this vector C. Would just be coming out towards you right? It's basically just going up like this, you're looking straight up at it. Okay let's do the next one here. So now we've got here is we've got this Y and Z. So again we have a different sort of orientation but it works the same. So we want to calculate the cross products. So we've got see the magnitude of C. Is a B. Sine Theta. Okay so the magnitude here is three magnitude here is eight. And then what about the angle that's between them? What do I plug into my sign here? Do I plug in 23? Do I plug in 67? Remember? It's the smallest angle between these two vectors here between them. So what you can do here is if they sort of both lie on the Y. Z. Plane like this. Remember that a straight line Is 180. So if you subtract 67 and 23, what you're gonna get out of this? Is that this is actually a perfect right angle. This is 90°. So we've got 90 degrees like this. So you're gonna stick in sine of 90. And what you're gonna get here is that this vector here is 24? Okay, so the sine of 90 is just one? Alright so where's the point? Well along the three D. View. What happens is we're gonna take your finger curl it, sorry point along a curl towards B. So if you do that, what happens is that your thumb is going to be pointing away from you? So the way this looks on a three dimensional sort of plane is that it be going backwards like this? Remember this is sort of like the front to back dimension. So it's gonna be pointing out like this. So this c vector here is gonna be 24 points sort of away from you like that and we would represent that again. It's just an X. A circle with an X. Like that. Okay now the last one here is we're going to do two and 4. So we got the magnitude which is just a be sine theta. Okay so now we've got the magnitude which is four magnitude which is two. And what about the sine of the angle between them? Well what happens is here, is that on a two dimensional sort of grid like this these these vectors are parallel. So what happens here is that the angle between them is zero? And if you plug this in, what happens is that the sine of zero is just zero. The whole thing just goes away. So in other words what happens is that the cross product is 0? If the vectors are parallel and this can happen one of two ways either the angle between them is zero or you can also have 1 80. If you were to plug in sign of 1 80 you're also gonna get zero. That happens if they're pointing away from each other. Alright, so that's it for this one guys, let me know if you have any questions.

Related Videos

Related Practice

Showing 1 of 8 videos