 Okay, this is going to be a super quick review of vectors. One of the important things is that in the curriculum that we're using, sometimes we represent vectors in a way that maybe wasn't the same as what you did before. So this may be review, but let's just start off with a vector. So here's my, let's say, x and y-axis. And here I have a vector, vector a, and it's a distance, I'm just going to say. And let's say this is three meters, and that's 20 degrees. I'm just, just making up stuff here, okay? How would we represent that vector in component notation? Well, in component notation, what I want to do is, is break this into a component along the x-axis, plus a component along the y-axis, plus a component along the z-axis. So you can see the right triangle right there. Now I can get the magnitude of the x-component, it's going to be, I'm going to write it like this, a equals 3 meters times cosine of 20 degrees, right? Because hypotenuse times cosine 20 will give me the adjacent side of that triangle. That's the right triangle, and put a brace like that. This is how we represent things. The y-component's going to be 3 meters sine of 20, and in this case, the z-component is zero, because it's not coming in and out of the board. So I'll just say zero meters. So in general, this is how we write a vector a equals a x, a y, a z. If we have a component already like that, and we want to find the magnitude, go backwards, I could just say the magnitude of a would be the square root of ax squared plus ay squared plus az squared. Just like the Pythagorean theorem, it works in three dimensions too. Okay, what else do we need to know about vectors? Let's say, and I didn't multiply that out because I'm lazy. Okay, let's just say that I have a different vector b, negative 1, 2, negative 3 meters. A vector has to have units. Sometimes we're lazy and I don't put them one there, but if you don't have units, then it's not a real thing, which sometimes can happen. But if it doesn't have units, that's important too. Each component has to have the same units. You can't have two meters that way and five newtons up. You can't do that. But what I can do is multiply, let's say, multiply or divide this. I could multiply this by, I'm trying to think of something. Let me say this is meters per second, just so it'll make more sense. So that's the velocity. Then I could find the momentum by multiplying by the mass. Let's say it has a mass of two kilograms. So p equals two kilograms times b. And so that would just be this scalar components multiplied by each of the components inside. So this would be negative 2, 4, negative 6 kilogram meters per second. If I want to add two vectors, let's say I have a vector c equals 0, 1, 2 kilogram meters per second. Then I can do c plus p. It has to have the same unit, so it would just be the sum of the components. So it's going to be 0 plus negative 2, 1 plus 4, and 2 plus negative 6. Now, another very important and useful thing is the unit vector. The unit vector is a vector along the axis of the vector, but it has a magnitude of one and no units. And that is confusing while we call it a unit vector, but it is the unit. Kind of think of it as you can multiply it by some quantity to make it into a vector. So let me just show this in general. Let me keep that same vector b. And I want to find the vector in the same direction as b, but with the magnitude of one and no units. So I call that b hat, and that's just going to be the vector b divided by the magnitude of b. So in this case, I could write that as negative 1, 2, negative 3 meters per second. And then this is going to be the square root of 1 squared plus 2 squared plus 3 squared. So this is going to be, that's 1 plus 4 is 5 plus 9 is 14, so square root of 14. But this is going to have units of meters per second, so those cancel. So I get negative 1 over the square root of 14, 2 over the square root of 14, negative 3 over the square root of 14. That's it. There are no units. Now, you could go ahead and find the magnitude of that vector, and you would see that it has the magnitude of 1. Okay, one last thing, and there's more with vectors, but I'm just trying to give you a quick review so you can watch things later. For both classes that I'm working on, you're going to do something like this. Especially in, let's say I have an electron right there and a proton right there. One of the important things I need to do is find a vector like this. I call that vector r. But normally I'm given a vector location, let's say from the origin. So let's call this r minus is a vector from the origin to the electron, and r plus is a vector from the origin to the proton. Now, how would I use those to find r? Well, one of the things that you can always say is you can actually remember that the change in position from here to there is like final minus initial, right? Whenever you change from one to another. So I'm going to end up at the plus and start at the minus so I could write r equals r plus minus r minus. And if I know the vector r and I know the vector r plus, then I can find r. That's something that you're going to do a lot in both classes. Okay, I think that's enough vectors right now, but I just wanted to put up something with vectors.