 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: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.