Welcome back, everyone. So, up to this point, we've spent a lot of time talking about vectors and what they look like. Now recall that visually, vectors are these arrows drawn in space, and we've talked about how these arrows can be stretched or shrunk, or combined with other arrows to create resultant vectors. Well, what we're going to be talking about in this video is position vectors and component form. And this might sound a bit random and confusing but don't sweat it because it turns out all we're really going to be learning about in this video is a simple way to write numbers that represent our vector. And I think you're going to find mathematically it's actually very intuitive. So without further ado, let's get right into things, because this is an important concept to understand.

Now let's say we have this vector, which we'll call vector \( \mathbf{v} \). As you can see, this is an arrow drawn in space. Now what we can do is represent this as a position vector, and a position vector is simply a vector where the initial point is drawn at the origin. So, if we want to draw vector \( \mathbf{v} \) as a position vector, I just need to relocate this so the initial point is at the origin. And that right there is the position vector, and that's all there is to it. It's just moving your vector to the origin of the graph.

Now the question becomes, how can we write this vector with some kind of numbers? Well, we can see that we have some numbers here on the graph, and we can use these numbers to figure out what our vector is. Because what we do is we represent these vectors using component form, where we have an x-component and a y-component. And all these components do is tell you the length of the vector in the x and y directions. So you can see in the x-direction, we need to go 3 units to the right, so we'd have 3. And then in the y-direction, we need to go 2 units up, so we'd have 2. So this vector is (3,2), and that's all there is to it. As you can see, it's really straightforward.

Now it turns out that there are also ways that you can represent these vectors if you don't have a position vector. So say that you were given some initial point of the vector, like a point right there, and then a terminal point over here. You could figure out what the vector is in component form by using this equation down here. And to really put this equation to use and understand it rather than just looking at it, let's actually try an example where we have to do this. So, in this example, we are told if a vector has initial (0,2,3) and a terminal (0,3,5), without drawing the vector, write the vector in component form. So we're not allowed to just graph this immediately and figure out what it looks like. What we need to do is use this equation to figure out what our vector is. But this equation is actually pretty simple to use. So all I need to do is recognize that our vector \( \mathbf{v} \) is going to be the difference in the x components, the difference in the y components.

So we can see that we have the final x minus the initial x, and then we're going to have the final y minus the initial y. Now I can see what these values are based on the points above. So if I go ahead and go to this first point, you see this first point is (0,2,3). I can see that the second point is (0,3,5). So I'm going to have for the x points is the difference between 3-2. So we're going to have 3 minus 2. And the reason that I put 3 first is that notice it's the final x minus the initial x. It's going to be 0.2 minus 0.1. So we have 3 minus 2, and then we're going to subtract the y-values. So the final y is 5, and the initial y is 3. This is what our vector is going to be. So we're going to have 3 minus 2 which is 1, and we're going to have 5 minus 3 which is 2. And that right there is the solution. That is vector \( \mathbf{v} \).

So this is how you can solve problems when you can't initially draw them or don't initially have some kind of graph of the vector. Now if we want to know what this vector looks like, we actually can use this graph just for reference. Well, I can see that our vector is (1,2). And if we draw this as a position vector, starting at the origin of our graph, we're going to go 1 to the right, and we're going to go 2 up. And that right there would be our vector \( \mathbf{v} \). So as you can see, whatever you're dealing with these types of vectors, you're going to have the x-component, which is how far we travel in the x-direction, and the y-component, which is how far we travel in the y-direction. And that's always going to be the case when using component form. So that is how you can represent vectors using numbers, and how you can draw position vectors which are at the origin of your graph. So hope you found this video helpful. Thanks for watching.