Question

Find the shortest distance between the lines `barr = (4hati - hatj) + λ(hati + 2hatj - 3hatk)` and `barr = (hati - hatj + 2hatk) + μ(hati + 4hatj - 5hatk)`

Answer 1

We know that the shortest distance between the skew lines

`barr = bara_1 + λbarb and barr = bara_2 + μbarb_2` is given by

d = `|((bara_2 - bara_1)*(barb_1 xx barb_2))/(|barb_1 xx barb_2|)|`

Here, `bara_1 = 4hati - hatj`,

`bara_2 = hati - hatj + 2hatk`,

`barb_1 = hati + 2hatj - 3hatk`,

`barb_2 = hati + 4hatj - 5hatk`

∴ `barb_1 xx barb_2 = |(hati, hatj, hatk),(1, 2, -3),(1, 4, -5)|`

= `(-10 + 12)hati - (-5 + 3)hatj + (4 - 2)hatk`

= `2hati + 2hatj + 2hatk`

and `bara_2 - bara_1 = (hati - hatj + 2hatk) - (4hati - hatj)`

= `-3hati + 2hatk`

∴ `(bara_2  - bara_1)*(barb_1 xx barb_2) = (-3hati + 2hatk)*(2hati + 2hatj + 2hatk)`

= –3(2) + 0(2) + 2(2)

= – 6 + 0 + 4

= –2

and `|barb_1 xx barb_2| = sqrt(2^2 + 2^2 + 2^2)`

= `sqrt(4 + 4 + 4)`

= `2sqrt(3)`

∴ Required shortest distance between the given lines

= `|(-2)/(2sqrt(3))|`

= `(1)/sqrt(3)"units"`.