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This is an example problem from the book vector mechanics for engineers. I had a doubt that how the position vectors for A and position vector B relative to A defined. That is the equation which is rounded in yellow in the second picture, there we should write the vertical component of of B/A and add it to position vector A. But we are adding magnitudes of lengths of vectors to obtain the length of wire. It is confusing. Please help

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In general you are right. It's not possible to casually add position vectors in different coordinate systems.

However in this case it is possible, because of the context. In order to help you understand it more intuitively, it might help you are considering that the constrain equation describes the length of the rope. And the values $x_A$ and $x_{B|A}$ are not vectors but they are scalar coordinates of the position vectors. The fact that the coordinate systems are selected in such a way that the rope is parallel, means that it is possible to use those coordinates directly in the constraint equation.

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  • $\begingroup$ So after differentiation we get velocities which will be magnitudes of the velocity vectors without the direction. Now I understand thank you. $\endgroup$ Feb 17, 2022 at 10:35
  • $\begingroup$ Again in this particular occassion (because the coordinates systems were selected smartly), the velocity magnitude that is calculated can be associated with the coordinate systems. E.g. you see that $v_A = 2\cdot v_{B|A}$ $\endgroup$
    – NMech
    Feb 17, 2022 at 10:44

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