A mountain has the shape of an elliptic paraboloid $$z=c−ax^2−by^2$$, where $a$, $b$, and $c$ are positive constants, $x$ and $y$ are the east-west and north-south map coordinates and $z$ is the altitude above sea level (measured in meters). An engineer wishes to build a railroad up the mountain. Straight up is much too steep. At the point $(1,1)$, in what directions may the track be laid so that it will be climbing with a $3%$ grade - that is, an angle whose tangent is $.03$.
My attempt: I rearranged it to be $$c = z+0.026x^2+0.015y^2$$ I found the gradient of this to be $(0.054x, 0.03y, 1)$. At $(1,1)$ it is simply $(0.054, 0.03, 1)$. Now I need to have some vector $\vec{u}$ so that $(0.054, 0.03, 1)$ dot product with $(u_1,u_2,u_3) = 0.03$.
I'm a little lost here, am I doing this correctly? Any push in the right direction would be appreciated