 Now we want to look at velocity and acceleration in three-dimensional space. We want to do that derivation for those two equations and the best way to do that though is to change your immediate frame to cylindrical coordinates. So let's have this. We have here our world coordinate system. Our world coordinate system. So we're going to make this the k hat direction. This the i hat direction, which means somewhere in that direction there will have our j hat. So I'm swinging from i to jk. That's the world. Here's my point a. Which is here in my cylindrical coordinate system. Let's view it something like this in any point there, any point there. And that's my point b. So I can express b as far as this a-frame is concerned. So once again, I'm going to remind you of the fact that r of b with respect to o, that's going to be r of a in o plus r of b in a. That's what we're gonna have. That was the normal thing. The only thing we're going to change now. We're going to express this vector a, b in cylindrical coordinates. And how would we do that? Now remember cylindrical coordinates. I'm going to have this r of b and a. I'm just going to write it down for you. We'll run that. That's going to be some r and r hat. Plus we're going to have a z in the k hat direction. And we're going to have some angle theta. Now that's a bit special there. Now just remind yourself of polar coordinates. Let's just do that. Let's just, or not polar coordinates, cylindrical coordinates. Let's just have the cylindrical coordinates. Remember with cylindrical coordinates the cylindrical coordinates. If I have this in my vector there, remember this is going to be whatever direction this is going to be. That is my r hat direction. So it's just going to have this length in that direction. Now so it's going to have some height in the k hat direction. In the k hat direction. And if I look at it from the top, if I look at it from the top, it's going to be at some angle there. But remember an angle does not fulfill, just a static angle. It does not fulfill the criteria for a vector. Because remember they don't commute. Remember first year of physics you learned if you first turned something in this direction and then this would end up being different from doing it this and then this direction. You end up with two different things. It's like putting on your underpants and your pants. Whatever you put on first it's going to make a difference. So this is not part, although to express the vector completely we've got to have this r in k hat and whatever angle it's standing at. So it's going to have this direction. It's going to have a certain height and it's going to point somewhere if I look from the middle from somewhere there. But where it points there is not of concern. It is not a vector. So this is what we end up with. So the first thing that we want really we want our dot over b in the o. And that is going to be our dot nothing is changed on a in o. I can just express it as such because this a is in the world frame. So if I take its first derivative I can just take the first derivative. But now I have to take the first derivative of these two. Most definitely r hat changes because it goes with the vector. If this point b is moving this point b is moving in three dimensional space. So in the i hat direction and the j hat direction and in the k hat direction. So this is not constant. k hat definitely is constant because k hat yeah small k hat equals uppercase. Uppercase and now k hat they point in the same direction both unit vectors. So that does not change. So this does not change as we change. As the position of b changes in the world frame. So I am going to take the product rule of these two. I can do the product rule of these two but the d dt of k hat is going to be zero. Let me write it out. So what are we going to be left with here? Plus we are going to have r dot r hat plus r and then the d dt of r hat plus I am going to have z dot k hat. And as I say plus let's just put it there z d dt of k hat. But k hat is a constant. The first derivative of constant is zero. So actually zero times z nothing gets left there. Now as with always we've got to remember what these are. You have to have them in your head. Remember we showed this many times before that the d dt of this vector r hat. This tangent vector is this going to be, is this going to have this theta dot theta hat direction. And d dt of theta, oops, if theta hat remember that's going to be negative omega dot in the r hat direction. Because if you view it from the top remember there's some angular velocity in that direction. Remember that we're going to have. That is, if that's the r direction you have some angular velocity theta dot, theta dot or omega. So it's going to be in that direction and if you take that derivative it's going to be in the negative r direction again. We've done those, we've done those quite a few times. So we're going to have that r dot here with respect to o is going to be r dot a with respect to o plus we're going to have r dot r hat plus. Remember for that we're going to have this. So we're going to have r theta dot theta hat plus z dot k hat. And that for three dimensional space expressing these in cylindrical coordinates will be our equation for velocity. And same is going to happen for acceleration. This is in the o frame so we can just take the derivative easily enough of that. We've got to do the product rule there and the product rule there. This is silly not there because once again k hat is going to, is not going to make a difference for us. So what are we going to have here? R double dot r hat plus r dot again with this theta dot theta hat. And now we've got to do this one on the scale on down here. So we're going to take the derivative of this with those two, those two plus the derivative of that and those two plus the derivative of that. So let's do that. So we're going to have plus r dot theta dot theta hat plus r theta double dot theta hat minus. And now we're going to have this minus r. There's a theta dot and another theta dot that's theta dot squared in the r hat direction for that. And then here we're just going to have plus z double dot k hat for that. Now we can just group the terms and we'll have acceleration of point B in the overall frame. That's going to be the acceleration of A and O frame plus we could put all these r hats together. There's an r hat and there's an r hat which is going to be r double dot minus r theta dot squared r hat. And then for the theta hats, for the theta hats here we're going to have plus. Now there's one and there's one exactly the same to r dot theta dot for those two. And our other one here is plus r theta double dot. It's all in the theta hat direction plus z double dot k hat. I hope you can see in the corner there. So here's our equation for velocity in three-dimensional space using cylindrical coordinates and for acceleration in three-dimensional space. Quite easy derivatives. Derivations.