Question

What will be the shortest distance between the lines, `vecr = (hati + 2hatj + hatk) + lambda(hati - hatj + hatk)` and `vecr = (2hati - hatj - hatk) + mu(2hati + hatj + 2hatk)`

Options

• `(3sqrt(2))/2`

• `2sqrt(29)`

• `3/sqrt(19)`

• `8/sqrt(29)`

Answer 1

`(3sqrt(2))/2`

Explanation:

The given lines are 

`vecr = hati - 2hatj - hatk + lambda(hati + hatj + hatk)`  .......(i)

`vecr = 2hati - hatj - hatk + mu(2hati + hatj + 2hatk)` ......(ii)

The shortest distance between the lines

`vecr = a_1 lambda vecb_1` and `vecr = a_2 lambda vecb_2` is given by

`d = |((veca_2 - veca_1)(vecb_1 - vecb_2))/(|vecb_1 - vecb_2|)|`

Comparing the equation (1) and (2) with `vecr = veca_1 + lambda vecb_1` and `vecr = veca_2 + muvecb_2` respectively, we have

`veca_1 = hati + 2hatj + hatk, veca_2 = 2hati - hatj - hatk`

`vecb_1 = hati - hatj + hatk` and `vecb_2 = 2hati - hatj + 2hatk`

Now, `veca_2 - veca_2 = hati - 3hatj - 2hatk` and `vecb_1 xx vecb_2`

= `|(hati, hatj, hatk),(1, -1, 1),(2, 1, 2)|`

= `(-2 - 1)hati - (2 - 2)hatj + (1 + 2)hatk`

= `-3hati - 0hatj + 3hatk`

∴ `(veca_2 - veca_1) * (vecb_1 xx vecb_2) = (hati - 3hatj - 2hatk) * (-3hati + 3hatk)`

= `(1)(-3) + (-33)(0) + (-2)(3)`

= `-3 - 6 = - 9` and `|vecb_1 xx vecb_2| = sqrt(9 + 9) = sqrt(18) = 3sqrt(2)`

∴  `d = |(-9)/(3sqrt(2))| = |(-3)/sqrt(2)| = (-3)/sqrt(2) = (3sqrt(2))/2`