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Why is the following implication true?

$${d\over dt}\vec v(t)=\vec n\times \vec v(t) \implies \vec n\cdot\vec v(t)=\vec n\cdot \vec v(0)$$

I think the result can be obtained by expressing $\vec v(t)$ in a Taylor series expansion in $t$. And from $\displaystyle {d\over dt}\vec v(t)=\vec n\times \vec v(t) $, we see that $\vec n\cdot \vec v^{(n)}(0)=0$ due to the $\times \vec n$ bit.

Is there a more direct way?

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up vote 1 down vote accepted

$$ \vec n\cdot\frac{d\vec v(t)}{dt}=\vec n\cdot(\vec n\times \vec v(t))=0 $$ then $$ \frac{d}{dt} \vec n\cdot\vec v(t)=0 $$ that means $\vec n\cdot\vec v(t)=constant$. The constant can be fixed through $\vec v(0)$ and you are done.

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Thank you, Jon! – Matthew 1629 Dec 9 '12 at 16:09

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