The distance between the lines r ̅ = ( ̂i + ̂j + ̂k) + λ (3 ̂i + ̂j +…

The given lines are r ̅  = ( ̂i +  ̂j  + ̂k) + λ (3 ̂i +  ̂j  + 2 ̂k)  and r ̅  = ( ̂2i + 2 ̂j  +3 ̂k) + μ (3 ̂i +  ̂j  +2 ̂k)

The given lines are parallel

a₁ ̅   =( ̂i +  ̂j  + ̂k),    a₂ ̅   = (2 ̂i + 2 ̂j  +3 ̂k) and b ̅   = (3 ̂i +  ̂j  +2 ̂k)

Therefore the distance between the two lines is given by 

d=  mod{ b ̅  x (a₂ ̅   - a₁ ̅  )/ mod(b) } 

⇒ d=  mod{  (3 ̂i +  ̂j  +2 ̂k) x (  ̂i +  ̂j  +2 ̂k)/ √9+1+4 }

⇒ d=  mod{ ( 0 ̂i + 4 ̂j  +2 ̂k)/ √14 }

⇒ d=  (√20/ √14 )