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Question
Find the shortest distance between the lines given by `vecr = (8 + 3lambdahati - (9 + 16lambda)hatj + (10 + 7lambda)hatk` and `vecr = 15hati + 29hatj + 5hatk + mu(3hati + 8hatj - 5hatk)`
Solution
We have `vecr = (8 + 3lambdahati - (9 + 16lambda)hatj + (10 + 7lambda)hatk`
= `8hati + 9hatj + 10hatk + lambda(3hati - 16hatj + 7hatk)`
⇒ `veca_1 = 8hati - 9hatj + 10hatk` and `vecb_1 = 3hati - 16hatj + 7hatk` .....(i)
Aslo `vecr = 15hati + 29hatj + 5hatk + mu(3hati + 8hatj - 5hatk)`
⇒ `veca_2 = 15hati + 29hatj + 5hatk` and `vecb_2 = 3hati + 8hatj - 5hatk` .....(ii)
Now, shortest distance between two lines is given by
`|((vecb_1 xx vecb_2) * (veca_2 - veca_2))/|vecb_1 xx vecb_2||`
∴ `vecb_1 xx vecb_2 = |(hati, hatj, hatk),(3, -16, 7),(3, 8, -5)|`
= `hati(80 - 56) - hatj(-15 - 21) + hatk(24 + 48)`
= `24hati + 36hatj + 72hatk`
⇒ `|vecb_1 xx vecb_2| = sqrt(24^2 + 36^2 + 72^2)`
= `12sqrt(2^2 + 3^2 + 6^2)`
= 84
Now `(veca_2 - veca_1) = (15 - 8)hati + (29 + 9)hatj + (5 - 10)hatk`
= `7hati + 38hatj - 5hatk`
∴ Shortest distance = `|((24hati + 36hatj + 72hatk) * (7hati + 38hatj - 5hatk))/84|`
= `|((2hati + 3hatj + 6hatk) * (7hati + 38hatj - 5hatk))/7|`
= `|(14 + 114 - 30)/7|`
= 14
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