I'm trying to understand how the Frenet frame is formed from the normal and tangent on a curve in $\mathbb{R}^3$.
For a curve $\gamma (s)$ in $\mathbb{R}^3$ parameterised by arc length let $T(s) = \gamma' (s)$ be the unit tangent.
From a previous questions, I understand that from $T . T = 1 $ we get $T . T' = 0$. Then this gives us that $T'(s) = \kappa (s) N(s)$ for $\kappa(s) \in \mathbb{R}$ for $N(s)$ the unit normal.
BUT surely in $\mathbb{R}^3$ there are infinitely many normals as the vector can just 'rotate' round the curve whilst remaining perpendicular to the tangent and so I can't my head around how we can deduce $T'(s) = \kappa N(s)$ just from that fact $T . T' = 0$ ?
Thanks!