Vector function tough question

Question

If a curve has the property that the position vector $\vec{r}(t)$ is always perpendicular to the tangent vector $\vec{r'}(t)$, how can I show that the curve lies on a sphere with center the origin?

This is a problem from J. Stewart's book, but I'm stuck, so any tip will be helpful

Thanks in advance

Answer 1

Hint: $\vec{r}(t)\cdot\vec{r}\,'(t) = \dfrac{1}{2}\dfrac{d}{dt}(\vec{r}(t)\cdot\vec{r}(t))$.