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Question
Find the shortest distance between the lines `barr = (4hati - hatj) + λ(hati + 2hatj - 3hatk)` and `barr = (hati - hatj + 2hatk) + μ(hati + 4hatj - 5hatk)`
Solution
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_2)*(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"`.
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