Let $S$ be the surface $−5x^2+4y^2+2z^2−4=−5$. The vector $\mathbf{n}$ is normal to $S$ at the point $(3,−3,2)$ and $\mathbf{n}\cdot\hat{\mathbf{i}}=−30$. Find $\mathbf{n}\cdot\hat{\mathbf{k}}$.
I found the normal to equal $(-30,-24,8)$ at the point $(3,-3,2)$. But I'm not sure what to do after that. What am I suppose to do with $\mathbf{n}\cdot\hat{\mathbf{i}}=−30$? And how will that help me find $\mathbf{n}\cdot\hat{\mathbf{k}}$ ?
Assuming you have the correct normal vector (I haven't checked the first part of the calculation) then:
$\vec{i}, \vec{j}, \vec{k}$ are the unit vectors in the x, y and z directions.
The question gives that $\vec{n}\cdot\vec{i} = -30$, which is consistent with your computed value for $\vec{n}$, i.e. it is the x component of your vector in Cartesian coordinates.
$\vec{n}\cdot\vec{k}$ is the z component of your vector, which is 8.