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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.

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Hint: So $\mathbf{b}(s)$ is a constant vector $\mathbf{b}(0).$ Now, consider the plane passing through $\mathbf{r}(0)$ with the normal vector $\mathbf{b}(0).$ Does the plane contain $\mathbf{r}(s)?$

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