# Frenet frame in $\mathbb{R}^2$

My questions are probably very elementary ones, but I couldn't answer them for myself and didn't find the answer online.

Let $\gamma$ be a $C^2$ curve in $\mathbb{R}^2$ parametrized by arclength, and let $\{T(s),N(s)\}$ be it's frenet frame:

• Why is it true that $2\langle\dot{T}(s), T(s)\rangle = \frac{d}{ds} \langle T(s),T(s)\rangle$?
• What does $det[\gamma'(t),\gamma''(t)]$ mean? I don't understand this notation.
• For the first: the inner product satisfies a product rule. For the second: it means the determinant of the matrix with those two vectors as its columns. – symplectomorphic Mar 14 '15 at 12:57
• I knew i was missing something obvious. Thanks! – Rui Carneiro Mar 14 '15 at 13:02

As symple said... Product rule $$\frac{d}{ds}\langle T(s), T(s)\rangle = \langle T'(s), T(s)\rangle + \langle T(s), T'(s)\rangle = 2 \langle T'(s), T(s)\rangle$$
Each of $\gamma'(t)$ and $\gamma''(t)$ is a $2$-dimensional vector. Put them together into a $2 \times 2$ matrix, and take the determinant of that. If this determinant is nonzero then $\gamma'(t)$ and $\gamma''(t)$ are not parallel ... And then the unit normal $N(s)$ can be defined.