The distance between the lines r ̅ = ( ̂i + 2 ̂j -4 ̂k) + λ (2 ̂i + 3…

The given lines are r ̅  = ( ̂i + 2 ̂j  -4 ̂k) + λ (2 ̂i + 3 ̂j  +6 ̂k)  and r ̅  = (3 ̂i + 3 ̂j  -5 ̂k) + μ (2 ̂i + 3 ̂j  +6 ̂k)

The given lines are parallel

a₁ ̅   = ( ̂i + 2 ̂j  -4 ̂k),    a₂ ̅   = (3 ̂i + 3 ̂j  -5 ̂k) and b ̅   = (2 ̂i + 3 ̂j  +6 ̂k)

Therefore the distance between the two lines is given by 

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

⇒ d=  mod{  (2 ̂i + 3 ̂j  +6 ̂k) x ( 2 ̂i +  ̂j  - ̂k)/ √4+9+36 }

⇒ d=  mod{ ( -9 ̂i + 14 ̂j  -4 ̂k)/ √4+9+36 }

⇒ d=  (√293/ √49 )

⇒ d=  (√293 /7 )