 So let's look at deriving the equations for velocity and acceleration, those vector equations, but in three-dimensional space. So we're going to start here. We're not really going to use any proper coding here. So I'm just going to import the images and the filter warnings. Let's look at an example. Now suppose we have an upright spinning cylinder there, which form the basis of a cylindrical coordinate system. Remember those? In this body coordinate system, which is an intermediate frame, any position B can now be expressed in terms of R, Theta and Z. Remember that? The origins of this intermediate frame is that A and A is a point in an inertial reference frame O. So let's have a look and see what that looks like. Here we go. So here's my body coordinate system. I have I hat, J hat, right-hand rule, K hat will be straight up. I have my point A, which is referencing my intermediate frame there. And in my intermediate frame I have this R hat direction, again in a K hat direction. And if I were to look from the top, some angle Theta. So that is my body coordinate system. So if I were just to look at position of B in O, it's given by R of A in O. So this is A in O plus B in A. But now I'm going to express B in A in a different form. I'm going to express it as R in the R hat direction, Z up in the K hat direction and Theta. But Theta you've got to be very careful about. That's looking from the top. It's going to make some angle, but Theta is not a vector. Remember that's not a vector. It doesn't commute, so you can't use that as a vector. So if I were to take the velocity, all I have to do is to remember which unit vectors are not constant. So when I do the product rule for differentiation that I don't, that I remember to do that derivative as well. That's one thing I do have to remember though that K hat and K hat is exactly the same vector. Lower case K hat, upper case K hat. So if I do take the derivative of K hat with respect to time, it's going to be zero because it is a constant. It does not change with respect to O there. The other thing that I have to remember is that the first derivative of R hat is Theta dot Theta hat and the first derivative with respect to T of Theta hat is negative Theta dot R hat. Remember that from down here. Just remind you of that. So at any time T, this is going to be a position vector and if it rotates in T plus Delta T, it's going to be there. So when this Delta T is very small, this arc length here is going to be my Delta R and remember that Delta R is going to be a Mega Delta T per second times second. So in the Theta hat direction because remember it's rotating in this upper direction. If I just solve for this Delta R over Delta T, I get a Mega Theta hat and if I take the limit as Delta T goes to zero of this, it becomes DDT of R hat which is a Mega Theta hat. So there's my R hat. It's rate of change as a body spins. It's going to be Theta dot Theta hat and the DDT of Theta hat here is going to be negative in the negative R direction. That's R. This is negative R. This is negative Omega Theta hat. We've seen those a hundred times. So taking the derivative for velocity, this first one there is also with respect to the world coordinate system so just taking its derivative is very simple. But now for this lot here I have to do the product rule so I'm going to get R dot R hat and then R DDT of R dot which I now know what it is and I get for this one, I don't have to do the product rule because K hat is a constant in the O frame. So that's going to be zero so I'm only left with a Z dot K hat. So just writing that out, velocity then is going to be like this or like this. It's going to be the velocity of A in the O frame plus R dot R hat is R Theta dot Theta hat and Z dot K hat. No problems there. I'm going to repeat the process here for acceleration remembering that the Z double dot K hat, no problems there. This one is going to be R double dot R hat and then R dot which becomes Theta dot Theta hat there and there's three terms there so I'll have to take the derivative of the first one, the derivative of the second one, the derivative of the third term so each individually which will leave me with one, two, three terms which will be R dot, there's my Theta double dot and yeah I've already taken the derivative of Theta hat which remembers negative omega dot R hat. So of all of that there's two identical terms there, I can just group the terms and I'm left with acceleration in three dimensional space using an intermediate difference frame expressed in cylindrical coordinates of the acceleration with A respect to O but then these three unit vectors in the K hat direction is the Z double dot but in the Theta hat direction I'm going to have two R dot omega plus R omega dot or R alpha and I'm going to have R double dot minus R Theta dot omega squared, excellent.