 So now that we've defined position in two-dimension, we can come back and define our displacement in two-dimension. So our displacement in one-dimension was the change in position, and that was given the symbol of delta x, the change in position. And it was the final x measurement minus the initial x measurement, or our final position minus our initial position. Now this is if I'm using a horizontal line. If I had a vertical line, then my displacement was the delta y, and I had y final minus y initial. Both of these cases were vectors because it mattered whether I ended up having a plus or a minus, whether I was moving to the left or the right or up or down. So in two-dimensions, it's still my change in position. But rather than using x or y, in two-dimensions, I use r as my position symbol. So our displacement is still the change in position, and it's still the final position minus the initial position. But now it's not just numbers we're subtracting here. This is a true vector subtraction, where these are arrows out at some direction out in space. So let's look at it from the equation point of view. If this is my equation for the change in position as the final position minus the initial position, I want to express that final position in terms of my ij equation. So my final position is equal to the final x position in the i hat direction, plus the final y component in the j hat direction. And a similar thing for my initial position. It's the initial x and y values that I'm using. So when I get ready to actually do my equation vector subtraction, what I see is in the i hat direction, it's x final minus x initial. And for the j hat direction, it's y final minus y initial. And that tells me what my final position minus our initial position is. But when I come back and look at what these things mean, that final position minus the initial position is my change in position. The x final minus x initial is a delta x. And the y final minus y initial is the delta y. So I see that my change in position has components of the change in x and the change in y, respectively. So this means if I use my standard vector notation, my change in position is the delta x i hat plus delta y j hat, the change in the horizontal position and the change in the vertical position. The magnitude is still figured out using the Pythagorean theorem, except for rather than it being just r, it's delta r delta x squared and delta y squared. And my direction theta is my inverse cosine of delta y over delta x. Let's look at an example here real quick. Let's say I could represent my initial position by 1 meter i hat plus 3 meters j hat. And let's say I had a final position, which was 4 meters i hat plus a negative 2 meters j hat. Well, when I do my vector subtraction, the x component becomes 4 meters minus 1 meter. And my j hat becomes minus 2 meters minus 3 meters, which gives me a total of 3 meters in the i hat direction and negative 5 meters in the j hat direction. Graphically, I could express my initial position that 1 meter and 3 meters in terms of a vector that moves right 1 meter and upwards 3 meters. So I put a dot there. My final position of 4 meters i hat plus a negative 2 meters j hat means I've got a vector down here in this direction, 4 to the right, 2 down. And again, I put a dot there. Now my change in position is moving from my initial position to my final position. And that's a vector which is moving 3 to the right and 5 down, just like I found when I looked at it purely from the equation point of view. So that's displacement in two dimensions, which is really a combination of displacement in one dimension and good use of our vector quantities.