Curve with unit speed and zero curvature lies on a straight line

Question

Prove that a curve with unit speed and zero curvature lies on a straight line.

I know that $\|\gamma''\|=\kappa = 0$, so $\langle \gamma'', \gamma'' \rangle =0$.

But where do I go from here?

Answer 1

As $\gamma$ is unit speed, the curvature of $\gamma$ is $\|\ddot \gamma\|$. Thus, $\ddot \gamma = 0$, so $\dot \gamma(s) = a \in \mathbb R^3$. Hence $\gamma(s) = as + b$. Therefore $\gamma$ lies on a straight line.