= Solution
Write the position <vector> as $\mathbf r(\rho,\phi,z)=(\rho\cos\phi,\rho\sin\phi,z)$. Its coordinate tangent <vectors> are
$$
\mathbf r_\rho=(\cos\phi,\sin\phi,0),\qquad
\mathbf r_\phi=(-\rho\sin\phi,\rho\cos\phi,0),\qquad
\mathbf r_z=(0,0,1).
$$
Their pairwise <dot products> vanish. Their lengths, the <scale factors of orthogonal coordinates>, are
$$
\boxed{h_\rho=1,\qquad h_\phi=\rho,\qquad h_z=1.}
$$
Thus the <cylindrical coordinate system> is orthogonal wherever it is a nonsingular coordinate chart. At $\rho=0$, the angular tangent <vector> vanishes and the azimuth is undefined; this is the usual <coordinate singularity> on the axis, not a failure of the <Euclidean geometry>.
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