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Question
Find the shortest distance between the lines whose vector equations are `vecr = (1-t)hati + (t - 2)hatj + (3 -2t)hatk` and `vecr = (s+1)hati + (2s + 1)hatk`.
Solution
Equations of given lines
`vecr = (1 - t)hati + (t - 2)hatj + (3 - 2t)hatk`
or `vecr = hati - 2hatj + 3hatk + t (-hati + hatj - 2hatk)`
and `vecr = (s + 1)hati + (2s - 1)hatj + (2s + 1)hatk`
or `vecr = hati - hatj + hatk + s (hati + 2hatj - 2hatk)`
Comparing the above equations with `vecr = vec(a_1) + λvec(b_1)` and `vecr = vec(a_2) +µvec(b_2)`,
`vec(a_1) = hati - 2hatj + 3hatk, vec(b_1) = - hati + hatj - 2hatk`
and `vec(a_2) = hati - hatj - hatk, vec(b_2) = hati + 2hatj - 2hatk`
∴ `vec(a_2) - vec(a_1) = (hati - hatj - hatk) - (hati - 2hatj + 3hatk) = hatj - 4hatk`
`vec(b_1) xx vec(b_2) = (- hati + hatj - 2hatk) xx (hati + 2hatj - 2hatk)`
= `|(hati, hatj, hatk), (-1, 1, -2), (1, 2, -2)|`
= `(-2 + 4)hati - (2 + 2)hatj + (-2 -1)hatk`
= `2hati - 4hati - 3hatk`
∴ `|vec(b_1) xx vec(b_2)|`
= `sqrt((2)^2 + (-4)^2 + (-3)^2)`
= `sqrt(4 + 16 + 9)`
= `sqrt29`
∴ Required minimum distance d = `|((vec(a_2) - vec(a_1)). (vec(b_1) xx vec(b_2)))/|vec(b_1) xx vec(b_2)||`
= `|((hatj - 4hatk). (2hati - 4hatj - 3hatk))/sqrt29|`
= `|(-4 + 12)/sqrt29|`
= `8/sqrt29`
Hence, the minimum distance between the given lines is `8/sqrt29`.
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