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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
Given lines are:
`barr = (4hati - hatj) + λ(hati + 2hatj - 3hatk)` and
`barr = (hati - hatj -2hatk) + μ(hati + 4hatj - 5hatk)`
Let `bar a_1 = 4 hat i - hat j, bar b_1 = hat i + 2 hat j - 3 hat k`
`bar a_2 = hat i - hat j - 2 hat k, bar b_2 = hat i + 4 hat j - 5 hat k`
`therefore bar a_2 - bar a_1 = -3 hat i - 2 hat k` and
`bar b_1 xx bar b_2 = |(i, j, k), (1, 2, -3), (1, 4, -5)|`
= `hat i (-10 + 12) - hat j (-5 + 3) + hat k (4 - 2)`
= `2 hat i + 2 hat j + 2 hat k`
`therefore |bar b_1 xx bar b_2| = sqrt(2^2 + 2^2 + 2^2)`
= `sqrt 12`
= `2 sqrt 3` and
`(bar a_2 - bar a_1) * (bar b_1 xx bar b_2)`
= `(-3 hat i - 2 hat k) * (2 hat i + 2 hat j + 2 hat k)`
= (−3) (2) + (0) (2) + (−2) (2)
= − 6 + 0 − 4
= − 10
∴ Shortest distance = `|((bar a_2 - bar a_1) * (bar b_1 xx bar b_2))/|bar b_1 xx bar b_2||`
= `|(- 10)/(2 sqrt 3)|`
= `5/sqrt 3` units
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