The cylindrical coordinates don't form a vector. They're just a triple of numbers that you can use to describe a point; they don't have properties of vectors such as linearity, e.g. the cylindrical coordinates of the midpoint between two points aren't the average of the cylindrical coordinates of the two points.
What your text is likely referring to by "cylindrical vector" is a representation of a vector in terms of the orthogonal basis given by the unit vectors along $\partial\vec r/\rho$, $\partial\vec r/\phi$ and $\partial\vec r/z$. This is a local basis that depends on $\vec r$, so you're right in saying that your book does a bad job explaining these things; it's ambiguous to ask you to express a vector in this way without specifying the point whose basis is to be used. I'll assume that they mean "express the vector from $C(3,2,-7)$ to $D(-1,-4,2)$ in cylindrical components at $C$". (You omitted the $7$, but it can be reconstructed from the lengths of the vectors.)
Since $\partial\vec r/\partial z$ is just the canonical unit vector in the $z$ direction, the component in that direction is just $2-(-7)=9$. To find the other two components, note that $\partial\vec r/\partial\rho$ points in the direction from the $z$ axis to the point, so the corresponding component is given by
$$
\left(\pmatrix{-1\\-4}-\pmatrix{3\\2}\right)\cdot\pmatrix{3\\2}\Big/\left|\pmatrix{3\\2}\right|=\frac{-24}{\sqrt{13}}\approx-6.66\;.
$$
Then the remaining component along $\partial\vec r/\partial \phi$ is determined up to a sign from the length to be
$$
\pm\sqrt{4^2+6^2-\frac{24^2}{13}}\approx\pm2.77\;,
$$
and the sign is determined by your convention for $\phi$.
Best Answer
A vector field is defined over a region in space $\mathbb{R}^3 :$ $(x,y,z)$ or $(r,\phi, z)$, whichever coordinate system you may choose to represent this space. Your vector $\vec{N}$ should be defined in this space at a position vector $\vec{r} = (x,y,z)$ or $(r,\phi, z)$. So you need to find
$\phi = \arctan \left( \frac{y}{x} \right)$
Simply put: A vector field without reference to its position makes no sense in a transformation of coordinates.