 Let's talk about vector components on a grid now. As a reminder, our vector components are the projections of that vector onto the x and y-axis. We can also think of those components as being the sides of a triangle formed by a particular vector. So here my vectors are in black, and the vector components are the red and the blue parts, both the x for the red and the y is the blue. If I've got simple vectors that have nice integer components, then I can draw those on a grid and then count the boxes to find the components. Now when we're counting boxes, we always start at the tail of the vector and move toward the head, regardless of which direction the vector points in, always start at the tail. So let's take a look at the x component first. So for x, we're counting how far to the right or left our vector moves. And if the vector moves to the right, that means I've got a positive x. And if my vector moves to the left, that means I've got a negative x component. And I would just count the boxes out. Here in this case, I'm moving one, two, three boxes to the right, so that's a plus three x. The y component is similar, but we're counting how far upwards or downwards we're moving, with up being positive y and down being negative y. Let's take a look at some examples of this so you can get a little bit of practice. Okay, so here I've got a vector that's overlaid on a faint graph paper here. For this particular vector, if I move to the right one box, that puts me directly underneath the arrowhead. And then I would move up one, two, three boxes to find my y component. So this vector is represented by the components of one for x and three for y, telling me how far I've moved. Now I can grab that particular arrow and move it around to any place I want. So let's say I moved it down here. And again, it doesn't matter where I start on my graph paper, all I need to know is count from the tail down to the head. In this case, I'm moving over two positions, so that would be a negative two for my x component. And down one would give me a negative for my y component. If I wanted to take that vector and pull it up, I've got a negative two for the x and a positive one for the y. So I would write out those components as negative two for the x and positive one for the y. So if you've got a vector and it's drawn on graph paper, you count the boxes to figure out the components. What if you're given a set of components and you're asked to draw the vector? Again, you count the boxes and you figure out how far you're moving left and right and how far you're moving up and down. So for this case, I've got a positive four for my x. So I want to start someplace. It doesn't matter where I start on my graph paper, but once I start there, I need to count over that I'm moving four spots to the right. But I still need to move downwards minus two, so that moves down two spaces. So I've gone over four, down two, and that represents a vector with these components. Now you could go through and try writing your components, sketching those out in pencil and putting down your four and then putting down your negative two, and that's going to show you where you have to start and where you have to end up. If you want a little bit more practice with these components on grid paper, there's a website, and I'll give you that address in a minute, that has a little program here that lets us grab vectors and move them around and adjust their size. In this case, as soon as I adjust the size of the vector, it tells me what those components are. So you can practice what that looks like. This website also gives the ability to change if you're showing the components, showing it either in the view of it projected against the particular axes or showing you the sides of the triangle. And so that can give you a view of how those components change if you've got different vectors at different angles. And if you want to go to that website, here it is. It's the P-H-E-T program through Colorado, and it's their vector addition simulation. We're going to see that more as we go through this semester. So if you've got simple vectors and they're shown on a grid or you're asked to put a simple vector onto a grid, remember to count the boxes right and left and up and down, and that's going to have their connection between your vectors and your components and your grid paper.