Answer
We find that $|\mathrm{r}(t)|$ is constant, therefore the points on the curve lie on a sphere around the origin.
(see proof in step-by-step)
Work Step by Step
If the position vector is perpendicular to the tangent vector, then
$\mathrm{r}(t)\cdot \mathrm{r}^{\prime}(t)=0$, for all t,
The result of the previous exercise gave us
$\displaystyle \frac{d}{dt}|\mathrm{r}(t)|=\frac{1}{|\mathrm{r}(t)|}\mathrm{r}(t)\cdot \mathrm{r}^{\prime}(t)$
So, in this case, where we have $\quad \displaystyle \frac{d}{dt}|\mathrm{r}(t)|=0,$
it follows that $|\mathrm{r}(t)|$ must be constant.
But, if it is constant, then the distance of any point on the curve to the origin is constant, meaning that all points on the curve lie on a sphere with origin as its center.
