Let $\alpha: I \subset \mathbb{R} \rightarrow \mathbb{R^3}$ be a differentiable curve. I'm in the process of proving that $$||\alpha(b) - \alpha(a)|| \leq L(\alpha)_a^b$$ being $L(\alpha)_a^b$ the length of the curve from a to be.
It says in my text book that, if $\alpha(b) \neq 0$ we can separate $\alpha'$ in $(t, b]$ as follows:
$$ \alpha'= \langle \alpha' , \frac{\alpha}{||\alpha||} \rangle \frac{\alpha}{||\alpha||} + \alpha^\perp $$
At first i thought of Gram–Schmidt process of orthogonalization but I don't recognize this decomposition and dont know why this equality holds.
If you have two vectors $v$ and $w$, with $w\ne0$, then$$v=\left\langle v,\frac w{\|w\|}\right\rangle\frac w{\|w\|}+\left(v-\left\langle v,\frac w{\|w\|}\right\rangle\frac w{\|w\|}\right).$$This equality expresses $v$ as the sum of a vector which is a multiple of $w$ with a vector which is orthogonal to $w$.