# Thread: Vector valued function problem

1. ## Vector valued function problem

"Show that, at a local max or min of $\displaystyle ||\vec{r}(t)||$, the vector $\displaystyle r'(t)$ is perpendicular to $\displaystyle \vec{r}(t)$"

I think that if $\displaystyle f(t) = ||\vec{r}(t)||^2$, then the local min/max of $\displaystyle f(x)$ should be the same as $\displaystyle ||\vec{r}(t)||$, but I am not quite sure how to proceed with the rest of the question after that or how to show that $\displaystyle r'(t)$ is perpendicular to $\displaystyle \vec{r}(t)$.

2. Here are a few relevant ideas:

1. $\displaystyle \|\mathbf{r}(t)\|^{2}=\mathbf{r}(t)\cdot\mathbf{r} (t)$

2. If $\displaystyle \|\mathbf{r}(t)\|$ has a max or min, then $\displaystyle \|\mathbf{r}(t)\|^{2}$ does as well. Proof: assume

$\displaystyle \dfrac{d}{dt}\|\mathbf{r}(t)\|=0.$

Then $\displaystyle \dfrac{d}{dt}\|\mathbf{r}(t)\|^{2}=2\|\mathbf{r}(t )\|\,\dfrac{d}{dt}\|\mathbf{r}(t)\|=0.$

QED.

3. The product rule for derivatives works for the dot product.

There are some hints for you.