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Hi I am a mathematics student with an interest in Physics. In our Physics elective our prof. said if $\vec r$ denotes the position vector then the velocity vector $\vec v = \vec {\dot r} $ is independent of $\vec r$. I don't understand what this means. Is it linear independence in the sense of linear algebra ? But in that case if $ \vec r (t) = t \hat i $ then $\vec r(t) $ and $ \vec v(t) = \hat i$ are always linearly dependent.

Kindly explain this concept to me.

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marked as duplicate by ACuriousMind Apr 14 at 18:34

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    $\begingroup$ If you are not asking about the independence of the generalized coordinates in Lagrangian mechanics (as the Newtonian mechanics tag suggests), please explain more of the context here (if possible, e.g. was this in the context of circular motion?) because otherwise it is impossible to tell what your professor may have meant. $\endgroup$ – ACuriousMind Apr 14 at 18:37
  • $\begingroup$ My question is more basic than that. What does independence mean in this context ? $\endgroup$ – Ignorant Mathematician Apr 14 at 18:39
  • $\begingroup$ @IgnorantMathematician If you take the derivative of the position vector, you can see that there are terms with $r$. And then depending on the type of the problem, you could evaluate those parameters. It could depend on $r$ in the end ... or it would not. $\endgroup$ – KV18 Apr 14 at 18:40
  • $\begingroup$ My point is that there is no meaning to "independence" in the generic Newtonian context. There is one in Lagrangian mechanics, and there may be one in specific Newtonian situations. $\endgroup$ – ACuriousMind Apr 14 at 18:58
  • $\begingroup$ Independence means that the speed of an object does not a priori depend on its position. It is the equations of motion that bring about such a relation. $\endgroup$ – my2cts Apr 14 at 21:56