Welcome to module 15 of 3 dimensional dynamics. Today's learning outcome will be to derive something that we're going to call rotational transformation matrices. And so, rotation matrices or rotation transformation matrices, are just going to be handy work savers to do the rotations quickly. And the last couple of modules we've gone through step by step the Eulerian angles, and rotation matrices will just save you a little time. And so, let's look at a generic rotation. We'll do an example with the Eulerian angles and the angular velocity in the next module, but for right now, let's take some frame F, and rotate it through a new frame, about the k-axis. And so we're going to move from frame F to frame B, frame F is the big I J K axis, frame B is going to be the little i j k axis. And so we're going to move through what I'm going to call a generic angle, or angular velocity, gamma dot, through an angle gamma. And I'm going to say gamma sub Z because it's about the k or the z-axis. And so, what we're going to do now is, we're going to express some arbitrary vector Q in both of those axes, in frame F and in frame B, and so. Frame F Q is going to have some component in the big I direction, some component in the big Y direction, and some component in the big Z direction. Similarly for the in the little i j k axis of frame B, you've got Q is equal to Q of X and the little i direction component and the little j direction in the component in the little k direction. So here's the arbitrary vector Q expressed in the F frame and expressed in the B frame. And we're going to do a mathematical operation. We're going to dot the first equation here by little i. And so each of the terms is dotted by little i. So when I dot my arbitrary vector Q by little i, that's going to give me the projection of Q on the x-axis. By definition of the the dot product. When I dot big I with little i, so here's big I and here's little i. The dot product, remember, is the cosine of the angle between those two axes or two vectors. And so the angle is cosine or, excuse me, gamma sub z. So this becomes cosine of gamma sub z. And then on this term I've got J, the big J dotted with little i. And so, this angle is 90 degrees. And so that means that this angle, between little i and big J, is 90 degrees minus gamma z. And so the cosine of 90 degrees minus gamma C, is the same as the sine of gamma sub z. And then finally we've got k dotted with i, big K is in the same direction as little k, so their 90 degrees out from little i cosine of 90 degrees is zero. And so here's my result, the x component for Q is Q x cosine gamma sub z plus QY sign gamma sub z. And so similar you can dot the first equation by J and then by K, and you'll get your Y component of Q, and you'll get your Z component of Q. Although for the Z component that's quite straightforward because little q sub Z and big Q sub Z since the K, we're rotating about the K axis, those are the same. So Q sub small Z and Q sub large Z are the same. Okay, so here's the results. When we go through these rotations, with the rotation matrices, it, it's a, it's a, it's easy way of expressing them, the rotated reference frame, is with a matrix form. And so I've written these three equations now in matrix form. Cosine, excuse me, Q sub X is equal to cosine gamma sub Z times Q sub big X. Little q sub Y is equal to minus sine of gamma sub Z times big Q sub X, and then finally Q sub Z is equal to little q sub, Q sub little z is the same as Q sub big Z. And so we've rotated around the Z axis from coordinates expressed in the big frame or the F frame to the B frame, and that's what we wanted to do. And so here's Q sub B, that's the expression of my vector in the B frame, here's Q sub F, that's my expression in the F frame, and so we call this matrix, the rotation transformation matrix. And so, rotational transformation matrix for rotation about the z-axis is shown here. Now you can do a similar approach for rotation about a generic x-axis and a generic y-axis. Now I'll leave that as an exercise on your own, but if you do that, you'll get your rotational transformation matrix generically about the x-axis, looks like this. And then about the y-axis, it looks like this, where I'm rotating through an arbitrary vector Q sub, or gamma sub x for the rotational transformation matrix around the, the x-axis. And about why for the y-axis. Okay, and so, you'll note that each T is called the rotational transformation matrix. But you'll also find that T transpose is the same as T inverse. and so, because that property is true, we call T or the rotational transformation matrix. It has the property which, that it is orthonormal, and so that's where we're going to pick up and we're going to use these rotation transformation matrix in the next module to solve a problem.