Step 1: Confirming the lines are parallel first:
Both direction vectors are literally \((2,3,6)\) — identical, not just proportional — so this is squarely the parallel-lines case, not the skew-lines case.
Step 2: Re-deriving the formula from perpendicular distance:
The shortest distance between two parallel lines equals the length of the component of the connecting vector \(\vec b_2-\vec b_1\) that is perpendicular to the common direction \(\vec d\); this perpendicular component's magnitude is exactly \(\dfrac{|(\vec b_2-\vec b_1)\times\vec d|}{|\vec d|}\) (area of the parallelogram divided by base length).
Step 3: Numeric evaluation:
With \(\vec b_2-\vec b_1=(2,1,-1)\) and \(\vec d=(2,3,6)\), the cross product is \((9,-14,4)\) of magnitude \(\sqrt{293}\), and \(|\vec d|=7\).
Final Answer:
Distance \(=\boxed{\dfrac{\sqrt{293}}{7}}\) units, matching the direct formula.