D/dt |r(t)| = (1/|r(t)|) r(t)dot r'(t)

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SUMMARY

The discussion focuses on proving the equation d/dt |r(t)| = (1/|r(t)|) r(t) · r'(t), where r(t) is a non-zero vector. The key insight is recognizing that |r(t)|^2 = r(t) · r(t) allows the application of the chain rule to differentiate both sides. Participants clarified that the derivative of the magnitude |r(t)| is not zero, as it can change over time depending on r(t). The dot product of r(t) and its derivative r'(t) is essential in this proof.

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Homework Statement


If vector r(t) is not 0, show that d/dt |r(t)| = (1/|r(t)|) r(t)dot r'(t).

The last part is the dot product of r(t) and r'(t).

Homework Equations



The hint given was that |r(t)|^2 = r(t) dot r(t)

The Attempt at a Solution



Not sure where to begin, but I thought that |r(t)| is the length or magnitude of the vector r(t), thus its derivative is zero. If this is correct, then I also need to prove that the right side = zero.
 
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Depending on what r(t) is, surely it's magnitude can change over time?

Can you take the derivative of both sides of the equation in the hint using the chain rule?
 
Thanks for the hint on the hint. So simple and I didn't see it before.
 

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