‫Welcome back.
‫Let's fill in the gaps now, if you know that both the axis overlap with each other or they're parallel
‫to each other, and that means that the body from Z equals the inertia from Z.
‫And to achieve that, you have to put zeros here and one here, then this is true.
‫Then the small Z influence, the inertial X and Y coordinates in terms of orientation, know the body
‫frame.
‫Z does not influence this angle here and therefore it does not influence the inertial X and Y coordinates.
‫That means that you should put zeros here and here because the body frame Z axis plays no role in contributing
‫to the inertial X and Y axis.
‫And so you have four gaps left.
‫The final four gaps are the same like in a two case.
‫Why?
‫Because the 2D case was simply a 3D case looked at from the top.
‫And now if you put it in a vector matrix form, then you get this thing here, this matrix, and there
‫you go.
‫You have your rotation matrix that rotates about the Z axis.
‫If you want to go the other way around, then you just take the inverse of that and that's it.
‫It would look like this.
‫You can also represent this concept like this.
‫You go from a body frame to an inertial frame using the rotation matrix that rotates about the Z axis.
‫And if you want to go back, you use its inverse matrix.
‫That's it for now.
‫Thank you very much.
‫And I'll see you in the next video.