So, for a quick review on vectors, remember a vector is a mathematical entity that represents both distance and magnitude or distance and direction. So, things like velocities where you have an object moving at some speed and some direction can be represented well with a vector. Geometrically, we often represent vectors using an arrow where the direction that the arrow points shows us direction and the length of the vector, the length of the arrow is the magnitude or length of the vector. So, if I call this vector V, then its magnitude is the length of that vector and then the direction is whatever direction it's pointing in. Now, if I have two vectors, vector V and vector U, there's a few things that we can do with that. One of them is scale the vectors. So, like if I wanted to look at the vector to U, that would be 2 of U and so that would be 1, 2 U. So, it points in the same direction as U, but it is twice as long. Likewise, negative U would be the opposite of U, so it would point in exactly the opposite direction. In this case, it was the same length because it was only scaled by 1, negative 1. Now, we can also combine vectors. So, if I want to take U plus V, I can do that geometrically by putting the vectors head to tail. So, here's the vector U. And then I add to that. So, from the end of that, I move by vector V. The idea here is you can think of them as like movements, like displacements. So, I start at home, I walk along the vector U, and then after that, I add to that a movement along vector V. And the resulting movement puts me from my home to where I ended up. And that's the vector U plus V. Now, we can subtract vectors by combining our scaling with our sums. So, if I want to find U minus V, I can take the vector U and combine it with the opposite of V, so I can draw a negative V vector. And then, from the start of the first to the end of the second, it would be the vector U minus V. And so, we can subtract vectors that way. Now, working with vectors totally geometrically is not very convenient. And so, oftentimes, we'll take our vector and resolve it into components. And so, we represent the vector as a combination of a horizontal piece and a vertical piece, the components of the vector. So, for example, let's say this vector had length 8 and an angle of 30 degrees. So, that gives us a magnitude and a direction. And so, this is the vector U. I could say that the magnitude of the vector U is 8. This double bar absolute value symbols are sometimes our vector norm symbols are used to represent the magnitude of the vector. Sometimes, you'll just see a single absolute value bar used, but this is more common. Then, we could find the components, the horizontal and vertical components here using trig. We can say this horizontal piece is going to be 8 cosine 30 degrees, this vertical bar 8 sine 30 degrees. And find that this horizontal part is going to be 4 root 3. This vertical part is going to be 4. And then, we need some way to represent that. And there's two common ways. One is using angle brackets, mainly used to differentiate a vector from a point. And so, this would be 4 root 3, 4. So, horizontal component and vertical component. Sometimes, you'll also see I and J used, where I vector is the vector with length 1 pointing to the right. And the vector J is the vector pointing up with length 1. And so, we could represent this as 4 root 3 in the I direction. So, 4 root 3 I and 4 in the up direction, so 4 J. And then, we've represented this vector using I's and J's. Now, one of the nice things about components of vectors is that it makes it very easy to combine vectors. So, if I had a vector U, which was 2, 3, and vector V, which is 5, negative 1, that is very easy for me to either scale or add these vectors. If I wanted to find 2U, for example, that I'm scaling this vector by 2 to scale the vector, I scale each component. So, I double the horizontal component, I double the vertical component. If instead, I wanted to add these vectors, so if I wanted to find U plus V, then I simply need to add the components. So, because I'm combining the horizontal movement from U with the horizontal movement from V, and similar for the vertical components. So, I can add the horizontal components, I can add the vertical components to find the sum of the vector. Likewise, I can do combinations of scaling and sums, and I can do subtraction of vectors the same way. So, that's a quick review of vectors.