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Question
By computing the shortest distance determine whether following lines intersect each other : `bar"r" = (hat"i" + hat"j" - hat"k") + lambda(2hat"i" - hat"j" + hat"k") and bar"r" (2hat"i" + 2hat"j" - 3hat"k") + mu(hat"i" + hat"j" - 2hat"k")`
Solution
Squares of lines are `bar"r" = (hat"i" + hat"j" - hat"k") + lambda(2hat"i" - hat"j" + hat"k")` and `bar"r" (2hat"i" + 2hat"j" - 3hat"k") + mu(hat"i" + hat"j" - 2hat"k")`
Here, `bar"a"_1 = hat"i" + hat"j" - hat"k", bar"b"_1 = 2hat"i" - hat"j" + hat"k"`,
`bar"a"_2 = 2hat"i" + 2hat"j" - 3hat"k", bar"b"_2 = hat"i" + hat"j" - 2hat"k"`
∴ `bar"a"_2 - bar"a"_1 = (2hat"i" + 2hat"j" - 3hat"k") - (hat"i" + hat"j" - hat"k")`
= `hat"i" + hat"j" - 2hat"k"`
∴ `bar"b"_1 xx bar"b"_2 = |(hat"i",hat"j", hat"k"),(2, -1, 1),(1, 1, -2)|`
= `(2 - 1)hat"i" - (-4 - 1)hat"j" + (2 + 1)hat"k"`
= `hat"i" + 5hat"j" + 3hat"k"`
`|bar"b"_1 xx bar"b"_2| = sqrt((1)^2 + 5^2 + 3^2)`
= `sqrt(1 + 25 + 9)`
= `sqrt(35)`
∴ `(bar"a"_2 - bar"a"_1).(bar"b"_1 xx bar"b"_2)`
= `(hat"i" + hat"j" - 2hat"k").(hat"i" + 5hat"j" + 3hat"k")`
= 1 + 5 – 6
= 0
Shortest distance between two lines
= `|((bar"a"_2 - bar"a"_1).(bar"b"_1 xx bar"b"_2))/|bar"b"_1 xx bar"b"_2||`.
= `|0/sqrt35|`
= 0
∴ Hence Given lines are interesting.
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