Question

Find the shortest distance between the lines L₁ & L₂ given below: 

L₁: The line passing through (2, −1, 1) and parallel to `x/1 = y/1 = z/3`

L₂:  `overset->r = hat i + (2mu + 1) hat j - (mu + 2) hat k`.

Answer 1

Given, L₁: The line passing through (2, −1, 1) and parallel to `x/1 = y/1 = z/3`

L₂:  `overset->r = hat i + (2mu + 1) hat j - (mu + 2) hat k`

Write the given equations of line in standard form

L₁:  `overset-> r = (2hati - hat j + hat k) + lambda(hat i = hat j + 3hatk)`       ...(i)

L₂:  `overset->r = (hat i + hat j - 2hatk) + mu(2hatj - hatk)`      ...(ii)

On comparing Eqs. (i) and (ii) with `overset->r = a_1 + lambdab_1 and overset->r = a_2 + lambdab_2` respectively, we get

`overset->a_1 = 2hati - hat j + hat k,  overset->b_1 = hat i + hat j + 3 hat k`

and `overset->a_2 = (hat i + hat j - 2hat k), overset->b_2 = 2hatj - hat k`

Clearly, `overset->b_1 xx overset->b_2 = |(hat i, hat j, hat k),(1, 1, 3),(0, 2, -1)|`

= `hat i(-1 -6) - hat j (-1-0) + hatk(2 - 0)`

= `-7hati + hat j + 2 hat k`

⇒ `|overset->b_1 xx overset->b_2| = |-7hat i + hat j + 2 hat k|`

= `sqrt((-7)^2 + (1)^2 + (2)^2)`

= `sqrt(49 + 1 + 4)`

= `sqrt54`

= `3sqrt6`

Now, `overset->a_2 - overset->a_1 = (hat i + hat j - 2hat k) - (2 hat i - hat j + hat k)`

= `- hat i + 2hat j - 3 hat k`

Required SD = `(|(overset->a_2 - overset->a_1)·(overset->b_1 xx overset->b_2)|)/|overset->b_1 xx overset->b_2|`

= `(|(-hat i + 2 hat j - 3 hat k)·(-7 hat i + hat j + 2 hat k)|)/(3 sqrt 6)`

= `|7 + 2 - 6|/(3 sqrt6)`

= `3/(3sqrt6)`

`1/sqrt6` unit