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I have the following multiple choice question with solution that I would like some clarification on. The question is as follows:

Suppose that $\vec{r}:\mathbb{R}\to\mathbb{R}^3$ is a curve given by $t\mapsto\vec{r}(t)$ and the dot product of the curve with its derivative is greater than zero. Which of the following can we conclude?

(a) $|\vec{r}|$ is increasing in $t$.

(b) $|\vec{r}'|$ is increasing in $t$.

(c) $|\vec{r}''|$ is increasing in $t$.

(d) None of the above.

The solution is said to be (a) with the following explanation, "Differentiate $f(t)=\langle \vec{r},\vec{r}'\rangle$ and note it is bigger than 0". This is unclear to me, any help would be appreciated as to why we can conclude (a) and the explanation given. Thanks.

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    $\begingroup$ I think they mean to consider $f(t) = \langle\vec{r}(t),\vec{r}(t)\rangle$ and differentiate that instead. Do you know how to differentiate an inner product? $\endgroup$
    – Joey Zou
    Oct 19, 2016 at 4:13

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As @Joey Zhou suggested $$ (|\vec{r}|^2)'=(\vec{r}•\vec{r})'=2\vec{r}'•\vec{r}>0 $$

So (a) is true.

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