I have proved that a planar curve of zero curvature is a straight line. It follows from the Frenet equations. But now I need to prove that if $\varkappa=0$, then the space curve $\mathbf{r}(t)$ is planar. From the condition and the Frenet equations it follows that $ \left\{ \begin{aligned} \frac{d}{ds}\mathbf{v}&=k(s)\mathbf{n}(s),\\ \frac{d}{ds}\mathbf{n}&=-k(s)\mathbf{v}(s),\\ \frac{d}{ds}\mathbf{b}&=0.\\ \end{aligned} \right. $
But how can be technically deduced from these equations that the curve is planar?
Update: from a related question planar curve if and only if torsion I have realized that I need to show that $(\mathbf{r}(t)-\mathbf{r}(t_0))\cdot\mathbf{b}(t)=0$ for any $t$ and some $t_0$. The question now is how to do that. I appreciate any help.