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‫Welcome back.

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‫Let's try to drive the rest of the two rotation mattresses now.

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‫Previously, we had derived a three day rotation matrix that rotated about the Z axis.

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‫Let's now look at the rotation about the Y axis.

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‫Here's the 3D frame.

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‫And now if you turn this frame in such a way that the Y axis stares right towards you, then in 2D you

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‫will see this picture.

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‫We describe the rotation about the Y axis with an angle theta.

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‫The rotation follows the right hand rule.

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‫As you can see, the body frame y and the inertial frame y, the overlap with each other to derive the

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‫3D rotation matrix about the Y axis.

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‫Let's rewrite the relationship of the inertial and body dimensions like this.

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‫First of all, you can see that the body frame y equals the inertia frame y.

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‫That means that we can put one here and zeroes here.

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‫Also the inertial X, Y and Z axis do not depend on the body frame Y axis either.

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‫Therefore we put zeroes here and here.

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‫However, they do depend on the body frame X, Y and Z axis and to fill in the rest of the blanks,

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‫us decompose the body frame X and Z vectors in the inertia frame.

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‫So in purple you will have the components of the body frame x axis in the direction of the inertia frame

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‫X and then the body frame x y component would be this one here.

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‫The body frame Z vector has this component here in green, in the inertia frame X direction, and it

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‫has another component in the inertia frame Z direction like this note is that here in this case, this

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‫is the positive direction for the inertia frame X dimension, and this is the positive direction for

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‫the inertia frame Z dimension.

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‫To get the horizontal part of the body frame x axis, you have to multiply it by cosine theta and so

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‫you put cosine theta here.

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‫And to get the horizontal part of the body from Z axis, you have to multiplied by sine theta.

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‫So you put sine feature here.

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‫Do you agree you have theta here and you need this component which is the same like this one, the vertical

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‫component of the body frame x axis is negative and you get it if you multiply small X by minus sine

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‫Seeta because it's negative, the vertical part of the body from Z axis points in the positive inertial

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‫frame Z direction, and you get it by multiplying small Z by cosine theta.

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‫So you put cosine theta here and there you go.

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‫You have filled in the blanks and now you know how to construct the rotation matrix.

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‫Right?

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‫It looks like this.

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‫You have the rotation matrix about the y axis and now try to do the same thing.

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‫But now your rotation matrix rotates about the X axis, the exact same logic.

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‫Only now you're rotating about the X axis and you'll see the solution in the next with you see there.