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‫Welcome back.

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‫Let's fill in the gaps now, if you know that both the axis overlap with each other or they're parallel

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‫to each other, and that means that the body from Z equals the inertia from Z.

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‫And to achieve that, you have to put zeros here and one here, then this is true.

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‫Then the small Z influence, the inertial X and Y coordinates in terms of orientation, know the body

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‫frame.

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‫Z does not influence this angle here and therefore it does not influence the inertial X and Y coordinates.

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‫That means that you should put zeros here and here because the body frame Z axis plays no role in contributing

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‫to the inertial X and Y axis.

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‫And so you have four gaps left.

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‫The final four gaps are the same like in a two case.

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‫Why?

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‫Because the 2D case was simply a 3D case looked at from the top.

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‫And now if you put it in a vector matrix form, then you get this thing here, this matrix, and there

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‫you go.

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‫You have your rotation matrix that rotates about the Z axis.

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‫If you want to go the other way around, then you just take the inverse of that and that's it.

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‫It would look like this.

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‫You can also represent this concept like this.

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‫You go from a body frame to an inertial frame using the rotation matrix that rotates about the Z axis.

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‫And if you want to go back, you use its inverse matrix.

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‫That's it for now.

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‫Thank you very much.

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‫And I'll see you in the next video.