When defining curvature my lecture notes say that for $\gamma (s)$ a curve parameterised by arc length.
Let $T(s) = \gamma' (s)$ be the unit tangent.
Let $N(s)$ be the unit normal.
Now $T . T = 1 $ and so $T . T' = 0$.
Thus we can write $T'(s) = \kappa (s) N(s)$ for $\kappa(s) \in \mathbb{R}$.
I'm probably missing something obvious but I don't understand the small stop from $T . T = 1 $ to $T . T' = 0$ and also how this then means we can derive $T'(s) = \kappa (s) N(s)$.
Any explanation would be appreciated!