I am reading about curves in the plane and curvature. In my literature it says that if $\alpha : (a,b) \rightarrow \mathbb{R}^2$ is a $C^2$-curve parameterized by arc-length, then $\{T(s), N(s)\}$ is an orthonormal basis for $\mathbb{R}^2$. That part I understand, but then it says that this implies that $\dot{T}(s)$ has the following expansion:
$$\dot{T}(s) = \langle\dot{T}(s),T(s)\rangle T(s) + \langle \dot{T}(s),N(s)\rangle N(s). $$
(Here $T(s)$ is the unit tangent, $N(s)$ is its normal and $\langle \cdot,\cdot \rangle$ is the inner product.)
I don't understand why this expansion is true. I'm guessing that we are multiplying $\langle \dot{T}(s),T(s) \rangle$ by $T(s)$ and by $N(s)$ because $T(s)$ and $N(s)$ are basis vectors, but I'm confused about why they are multiplied by $\langle \dot{T}(s),T(s)\rangle$ specifically.
If $\{w_1,w_2\}$ is an orthonormal basis of $\mathbb R^2$, then$$(\forall v\in\mathbb R^2):v=\langle v,w_1\rangle w_1+\langle v,w_2\rangle w_2.$$This is all that is being used here.