‫Welcome back.
‫Let's try to drive the rest of the two rotation mattresses now.
‫Previously, we had derived a three day rotation matrix that rotated about the Z axis.
‫Let's now look at the rotation about the Y axis.
‫Here's the 3D frame.
‫And now if you turn this frame in such a way that the Y axis stares right towards you, then in 2D you
‫will see this picture.
‫We describe the rotation about the Y axis with an angle theta.
‫The rotation follows the right hand rule.
‫As you can see, the body frame y and the inertial frame y, the overlap with each other to derive the
‫3D rotation matrix about the Y axis.
‫Let's rewrite the relationship of the inertial and body dimensions like this.
‫First of all, you can see that the body frame y equals the inertia frame y.
‫That means that we can put one here and zeroes here.
‫Also the inertial X, Y and Z axis do not depend on the body frame Y axis either.
‫Therefore we put zeroes here and here.
‫However, they do depend on the body frame X, Y and Z axis and to fill in the rest of the blanks,
‫us decompose the body frame X and Z vectors in the inertia frame.
‫So in purple you will have the components of the body frame x axis in the direction of the inertia frame
‫X and then the body frame x y component would be this one here.
‫The body frame Z vector has this component here in green, in the inertia frame X direction, and it
‫has another component in the inertia frame Z direction like this note is that here in this case, this
‫is the positive direction for the inertia frame X dimension, and this is the positive direction for
‫the inertia frame Z dimension.
‫To get the horizontal part of the body frame x axis, you have to multiply it by cosine theta and so
‫you put cosine theta here.
‫And to get the horizontal part of the body from Z axis, you have to multiplied by sine theta.
‫So you put sine feature here.
‫Do you agree you have theta here and you need this component which is the same like this one, the vertical
‫component of the body frame x axis is negative and you get it if you multiply small X by minus sine
‫Seeta because it's negative, the vertical part of the body from Z axis points in the positive inertial
‫frame Z direction, and you get it by multiplying small Z by cosine theta.
‫So you put cosine theta here and there you go.
‫You have filled in the blanks and now you know how to construct the rotation matrix.
‫Right?
‫It looks like this.
‫You have the rotation matrix about the y axis and now try to do the same thing.
‫But now your rotation matrix rotates about the X axis, the exact same logic.
‫Only now you're rotating about the X axis and you'll see the solution in the next with you see there.