Question

Find the shortest distance between the lines `vec r = hat i + 2hat j + 3 hat k +  lambda(2 hat i +  3hatj +  4hatk)` and `vec r =  2hat i +  4 hat j + 5 hat k +  mu (4hat i + 6 hat j +  8 hat k)`

Answer 1

The given lines are 

`vec r = (hat i +  2 hat j + 3 hatk ) + lambda(2 hati +  3 hat j +  4 hat k)`

and 

`vec r = (2 hat i + 4 hat j +  5 hatk) +  2 mu(2 hati +  3 hatj +  4 hat k)`

or 

`vec r = (2 hat i + 4 hat j + 5 hat k) + mu^'(2 hat i +  3 hat j + 4 hat k)`

replacing `2mu` by  asingle parameter `mu'`

These two lines pass through the point A and B having position vectors `vec (a_1) = hat i +  2 hat j + 3 hat k` and `vec(a_2) = 2hati + 4 hatj +5 hat k`

`S.D. = |(vec(a_2) - vec(a_1))xx vec b|/|vec b|`

Here `(vec (a_2) - vec (a_1)) = (2hati + 4 hatj + 5 hatk) - (hat i + 2hatj + 3hatk) = hati + 2hatj + 2 hatk`

`:. (vec(a_2) - vec(a_1)) xx vecb = (hat i +  2hat j +  2 hatk) xx (2 hat i + 3 hatj + 4 hatk)`

`= |(hati, hatj, hatk),(1,2,2),(2,3,4)| = (8 - 6)hat i - (4 - 4) hatj + (3 - 4)hat k = 2hat i - 0hatj - hatk`

`:. |(vec(a_2) - vec(a_1))xx vecb| = sqrt((2)^2 + 0^2 + (-1)^2) = sqrt5` and `|vec b| = sqrt(4 + 9 + 16) = sqrt29`

Substituting these values in the formula for S.D. we have S.D. = `sqrt5/sqrt29 = sqrt(5/29)` units