In the following exercise, solution of the first exercise, why does $\cos{\theta} = \langle \gamma', e_1\rangle$? From there, it is easy, but I do not understand why $\cos{\theta} = \langle \gamma', e_1\rangle$ is true. My attempt involving that $\cos{\theta} = F / \sqrt{EG}$, but I did not arrive at this identity.
In general, for any two vectors $v$, $w$ at some point, we have $$ \langle v,w\rangle = \|v\|\|w\|\cos \theta$$ where $\theta$ is the angle between them. Since $v=\gamma'$ and $w=\mathbf{e}_1$ have unit length, one finds $\cos \theta= \langle \gamma',\mathbf{e}_1 \rangle$.