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I'm trying to express the velocity of a point P by using a rotation matrix between two reference frames (see attached file please).

Notation: the vertical bar with $O_1$ or $O_2$ means that the vector's components are expressed in the $O_1$ or $O_2$ reference frame. The dot over a vector means derivative with respect to time of that vector.

Is my procedure correct? In the last equation I used the Poisson's formulas to express the derivative of the rotation matrix.

Thank you so much.

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