Hey, guys. So in a previous video, I covered that there were 2 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 going to be using this a lot sometimes in physics. So let's go ahead and check this out here, and we're going to do a bunch of examples.

Now before we begin, I just want to briefly recap dot products. So imagine I had these 2 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 3 and this 4 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 abcosθ. So, if we just do this here, a dot b was a b cos. So in other words, it was just the magnitude of A, which is 3, magnitude of B, which is 4 times the cosine of the angle between them, which is 60 degrees. What you get out of this is you just get the number 6. And what the 6 really just represents here is it just represents the multiplication of parallel components of A and B. So, for example, this B vector here has a component that lies in the same direction parallel to A, and if you work this out, you're going to get the B x component is 2. So if you multiply these two components here that are parallel, you end up just getting 6 out of it.

Alright. So that's the scalar product. Now we're going to do the same thing. We're going to multiply these using the vector product 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 going to get a new vector. That's why we call it the vector product, and this vector here is going to be called C. Alright? So with the scalar product, you multiply them, you just get a number. With the vector product, we're going to multiply these 2 vectors and we're going to get a third one out of it called C. Now 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 going to use to calculate the magnitude is going to be very similar to the dot product. We're going to use a×bsinθ. So before we used the cosine, but now we're going to use the sine. So it's the magnitude of A, magnitude of B times the sine of the angle, and this angle here θ represents the smallest of the angles between A and B. So in other words, there are 2 ways to represent this angle. You can go all the way around, but that's not the smallest angle, so we're just going to use the 60. Alright? So if we do the vector product of these two vectors, what you're going to get is a third vector, which is C, and the magnitude of the C vector is just going to be a b times sine of θ. So we're going to do the same thing we did over here. This is going to be 3 and 4, 3 times 4, except now we're going to do sine of 60. And if you work this out, what you're going to get here is 10.4. The magnitude of this vector is not going to be 6. It's going to be 10.4 because now we're using the sine. Okay? But this is a vector. So remember the vectors have both magnitude and direction. Where does this 10.4 vector point? And to do that, we're going to 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 going to 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. Alright? So here's how the right-hand rule works. What you're going to do is you're going to take your hand and you're going to point your fingers along the first vector, always. You're always going to point along the first vector which in our case is A. So we're going to point our fingers along A like this. Okay? Now what you want to do is you want to curl your fingers towards the second vector. So you're going to curl them towards B. So in other words, I'm going to curl my fingers up like this in the direction where B is. And once 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 going to take your hand, point at 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 an xy-plane is we just draw a little circle with a dot going through this. This symbol here, the circle with 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 you're curling, first to you're pointing your fingers along the first and then you're curling towards the second one.