I am slightly confused by the following curve $\gamma(t) = (e^t,0,0)$ in $\mathbb{R}^3$. Its curvature, defined as $$ \kappa(t) = \frac{\|\dot \gamma(t) \times \ddot \gamma(t)\|}{\|\dot \gamma(t)\|^3} $$ vanishes everywhere, yet if I think about the curve geometrically it does have curvature. What am I missing ? Do I need a unit-speed parametrization to obtain the correct result?
Many thanks for your help!