Question

Find the shortest distance between the lines l₁ and l₂ whose vector equations are `overset->r = (-hati -hatj -hatk) + lambda(7hati - 6hatj + hatk)  "and"  overset->r = (3hati + 5hatj + 7hatk) + mu(hati - 2hatj +hatk)` where λ and μ are parameters.

Answer 1

l₁: `overset->r = (-hati -hatj -hatk) + lambda(7hati - 6hatj + hatk)`

`overset->a_1 = -hati -hatj -hatk`

`overset->b_1 = 7hati - 6hatj + hatk`

l₂: `overset->r = (3hati + 5hatj + 7hatk) + mu(hati - 2hatj +hatk)`

`overset->a_2 = 3hati + 5hatj + 7hatk`

`overset->b_2 = hati - 2hatj +hatk`

∵ `a_1/a_2 ≠ b_1/b_2 ≠ c_1/c_2`

∴ l₁ and l₂ are not parallel.

∴ l₁ and l₂ are skew lines.

`overset->b_1 xx overset->b_2 = |(hati,hatj,hatk),(7,-6,1),(1,-2,1)|`

= `hati [-6 + 2] - hatj [7 - 1] + hatk [-14 + 6]`

= `-4hati - 6hatj - 8hatk`

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

`(a_2 - a_1) = 4hati + 6hatj + 8hatk = 2[2hati + 3hatj + 4hatk]`

For skew lines,

D = `|((a_2 - a_1).(overset->b_1 xx overset->b_2))/|(overset->b_1 xx overset->b_2)||`

= `|((2[2hati + 3hatj + 4hatk]).(-2)(2hati + 3hatj + 4hatk))/sqrt((-2)^2(4 + 9 + 16))|`

= `|(2(4 + 9 + 16) (-2))/(2sqrt29)|`

= `(2 xx 29)/sqrt29`

D = `2sqrt29` unit