Give any arclength parametrized space curve $\alpha(s)$ (where space curve just means its codomain is $\mathbb R^3$), I want to show the following inequality:
$$\lVert \alpha(s) \rVert ≥ \lvert \alpha(s) \cdot N(s) \rvert$$
where $N(s)$ is the principal normal vector of $\alpha$.
I'm pretty sure this is true, as I can't think of any counterexamples, but I can't come up with a proof. So how can I do prove this? Or alternatively, if it's not true, under what conditions is it true?
Hint Recall the dot product formula $${\bf x} \cdot {\bf y} = ||{\bf x}||\,||{\bf y}|| \cos \theta ,$$ where $\theta$ is the angle between ${\bf x}$ and ${\bf y}$.