= Solution
Use direction $\mathbf u=(-3,-2,-1)$ and displacement $\mathbf w=(1,2,3)-(6,5,4)=(-5,-3,-1)$. The <point-to-line distance> is the height of the <parallelogram> spanned by these vectors:
$$
\boxed{D=\frac{|\mathbf w\times\mathbf u|}{|\mathbf u|}=\frac{|(1,-2,1)|}{\sqrt{14}}=\frac{\sqrt{21}}7.}
$$
Equivalently the closest line parameter is $t=(\mathbf w\cdot\mathbf u)/|\mathbf u|^2=11/7$, giving the foot $(9/7,13/7,17/7)$. Its separation from the point is <orthogonal> to $\mathbf u$ and has the same length.
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