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Question
Find the shortest distance between the lines given by `vec"r" = (8 + 3lambdahat"i" - (9 + 16lambda)hat"j" + (10 + 7lambda)hat"k"` and `vec"r" = 15hat"i" + 29hat"j" + 5hat"k" + mu(3hat"i" + 8hat"j" - 5hat"k")`
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Solution
Given equations of lines are
`vec"r" = (8 + 3lambdahat"i" - (9 + 16lambda)hat"j" + (10 + 7lambda)hat"k"` ......(i)
And `vec"r" = 15hat"i" + 29hat"j" + 5hat"k" + mu(3hat"i" + 8hat"j" - 5hat"k")` ......(ii)
Equation (i) can be re-written as
`vec"r" = 8hat"i" - 9hat"j" + 10hat"k" + lambda(3hat"i" - 16hat"j" + 7hat"k")` ......(iii)
Here, `vec"a"_1 = 8hat"i" - 9hat"j" + 10hat"k"` and `vec"a"_2 - 15hat"i" + 29hat"j" + 5hat"k"`
`vec"b"_1 = 3hat"i" - 16hat"j" + 7hat"k"` and `vec"b"_2 = 3hat"i" + 8hat"j" - 5hat"k"`
`vec"a"_2 - vec"a"_1 = 7hat"i" + 38hat"j" - 5hat"k"`
`vec"b"_1 xx vec"b"_2 = |(hat"i", hat"j", hat"k"),(3, -16, 7),(3, 8, -5)|`
= `hat"i"(80 - 56) - hat"j"(-15 - 21) + hat"k"(24 + 48)`
= `24hat"i" + 36hat"j" + 72hat"k"`
∴ Shortest distance, SD = `|((vec"a"_2 - vec"a"_1)*(vec"b"_1 xx vec"b"_2))/(|vec"b"_1 xx vec"b"_2|)|`
= `|((7hat"i" + 38hat"j" - 5hat"k")*(24hat"i" + 36hat"j" + 72hat"k"))/sqrt((24)^2 + (36)^2 + (72)^2)|`
= `|(168 + 1368 + 360)/sqrt(576 + 1296 + 5184)|`
= `|(168 + 1008)/sqrt(7056)|`
= 14 units.
Hence, the required distance is 14 units.
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