 Now let's look at some of our other motion equations in two-dimension. As a reminder, these are some of the one-dimensional motion equations I had for general motion in one-dimension. I could define my average velocity and my average acceleration, my instantaneous velocity, and my instantaneous acceleration. The instantaneous versions use the derivatives, where the average versions just use the delta x and delta t versions. And because I was only working in one-dimension, I just used delta x as my position vector when I was doing my velocity. If I want to work in two-dimensions, let's look at each one of these equations separately. So in two-dimensions, my average velocity vector is the displacement vector, which is now delta r over delta t. Because this delta r was a vector, I could break that down into its components to find that my average velocity vector has an x component and a y component, where the x component is delta x over delta t, and the y component is delta y over delta t. I could also write this out explicitly by saying my delta x over delta t is the x component of the average velocity, and the delta y over delta t is the y component of the average velocity. If I use derivatives, it works out the same sort of way. It's the position vector here. So I've got a dx dt and a dy dt giving me my i and j components. And I could write this out in terms of the dx dt is the x component of the velocity. And the dy dt is the y component of the velocity. Average acceleration works the same way, where now I'm using the change in the velocity vector. And so I've got a change in the vx component and a change in the vy component, giving me my two components for the average acceleration. And that means my delta vx dt, the change in the horizontal velocity divided by the change in time, is my x component of the average acceleration. And the change in the vertical velocity over the change in time span gives me the y component of the average velocity. It's a little bit neater again when I move to the derivative version here, but it's the derivative of the vx component and the derivative of the vy component. And that gives me my ax and my ay. And that's my two-dimensional motion equations.