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