Official solution
\[{L_1}:{{x + 7} \over 6} = {{y - 6} \over 7} = {{z - 0} \over 1}\]
Any point on it \[{\overrightarrow a _1}( - 7,6,0)\] and L1 is parallel to \[{\overrightarrow b _1}( - 6,7,1)\]
\[{L_2}:{{x - 7} \over { - 2}} = {{y - 2} \over 1} = {{z - 6} \over 1}\]
Any point on it \[{\overrightarrow a _2}(7,2,6)\] and L2 is parallel to \[{\overrightarrow b _2}( - 2,1,1)\]
Shortest distance between L1 and L2
\[ = \left| {{{({{\overrightarrow a }_2} - {{\overrightarrow a }_1})\,.\,({{\overrightarrow b }_1} \times {{\overrightarrow b }_2})} \over {\left| {{{\overrightarrow b }_1} \times {{\overrightarrow b }_2}} \right|}}} \right| = \left| {{{( - 14,4, - 6)\,.\,(3,2,4)} \over {\sqrt {9 + 4 + 16} }}} \right|\]
\[ = 2\sqrt {29} \].