5. Let $f(x,y)=x^2-e^{xy^2}$ and the surface $S$ be the graph of the function $z=f(x,y)$. Let $P=(1,0,0)\in S$ and $p=(1,0)$ in the $xy$-plane.

(a) $(10\%)$ If the unit vector $\mathbf{u}$ (in the $xy$-plane) at $p$ is the direction (among all directions at $p$) along which the height (i.e. the value of $z$) of $S$ increases most rapidly, then $\mathbf{u}=\underline{\quad(7)\quad}$.

(b) $(10\%)$ Write $H$ for the plane $x+2y+3z=1$ and the curve $C$ for the intersection $S\cap H$. Let $L$ be the tangent line to $C$ at $P$ and $N$ be the plane perpendicular to $L$ at $P$. Then the equation of $N$ is $\underline{\quad(8)\quad}$.