Let $\Phi$ be a surface in $\mathbb{R}^3$ with parameter domain $K\subset\mathbb{R}^2$ and let $\gamma:[a,b]\to K$ be a $\mathcal{C}^1$-curve. Also let $\alpha=\Phi\circ\gamma$. Prove that $\alpha^{\prime}(t)$ is orthogonal to $N(\gamma(t))$ for each $t\in[a,b]$, where $N(\gamma(t))$ denotes the normal vector to $\gamma$ at t.
I don't even know where to begin to tackle this problem. Any help is appreciated.
By definition: the un-normalized normal is
$$N(\gamma(t))=\gamma''-\left\langle \gamma'',\,\frac{\gamma'}{||\gamma'||}\right\rangle\frac{\gamma'}{||\gamma'||}=\gamma''-\frac1{||\gamma'||^2}\langle\gamma'',\gamma'\rangle\gamma'$$
Since $\;\alpha'=\phi'(\gamma)\gamma'\;$ , we get:
$$\left\langle\,\alpha',N\,\right\rangle=\phi'\langle\gamma',\gamma''\rangle-\frac{\phi'}{\rlap{\;\;\;\color{red}/}||\gamma'||^2}\rlap{\;\;\;\;\;\;\color{red}/}\left\langle\,\gamma',\,\gamma'\,\right\rangle\langle\gamma'',\gamma'\rangle=\langle,\rangle=\phi'\left(\langle\gamma',\gamma''\rangle-\langle\gamma'',\gamma'\rangle\right)=0$$