QJEEM22Maa427: JEE Main 2022 Mathematics - Class 12 JEEM22Maa427 Solution: The shortest distance between the lines and is :

${L_1}:{{x + 7} \over 6} = {{y - 6} \over 7} = {{z - 0} \over 1}$

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

$= \left| {{{({{\overrightarrow a }_2} - {{\overrightarrow a }_1})\,.\,({{\overrightarrow b }_1} \times {{\overrightarrow b }_2})} \over {\left| {{{\overrightarrow b }_1} \times {{\overrightarrow b }_2}} \right|}}} \right| = \left| {{{( - 14,4, - 6)\,.\,(3,2,4)} \over {\sqrt {9 + 4 + 16} }}} \right|$

$= 2\sqrt {29}$ .