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Question
Find the shortest distance between the lines `vec r = hat i + 2hat j + 3 hat k + lambda(2 hat i + 3hatj + 4hatk)` and `vec r = 2hat i + 4 hat j + 5 hat k + mu (4hat i + 6 hat j + 8 hat k)`
Solution
The given lines are
`vec r = (hat i + 2 hat j + 3 hatk ) + lambda(2 hati + 3 hat j + 4 hat k)`
and
`vec r = (2 hat i + 4 hat j + 5 hatk) + 2 mu(2 hati + 3 hatj + 4 hat k)`
or
`vec r = (2 hat i + 4 hat j + 5 hat k) + mu^'(2 hat i + 3 hat j + 4 hat k)`
replacing `2mu` by asingle parameter `mu'`
These two lines pass through the point A and B having position vectors `vec (a_1) = hat i + 2 hat j + 3 hat k` and `vec(a_2) = 2hati + 4 hatj +5 hat k`
`S.D. = |(vec(a_2) - vec(a_1))xx vec b|/|vec b|`
Here `(vec (a_2) - vec (a_1)) = (2hati + 4 hatj + 5 hatk) - (hat i + 2hatj + 3hatk) = hati + 2hatj + 2 hatk`
`:. (vec(a_2) - vec(a_1)) xx vecb = (hat i + 2hat j + 2 hatk) xx (2 hat i + 3 hatj + 4 hatk)`
`= |(hati, hatj, hatk),(1,2,2),(2,3,4)| = (8 - 6)hat i - (4 - 4) hatj + (3 - 4)hat k = 2hat i - 0hatj - hatk`
`:. |(vec(a_2) - vec(a_1))xx vecb| = sqrt((2)^2 + 0^2 + (-1)^2) = sqrt5` and `|vec b| = sqrt(4 + 9 + 16) = sqrt29`
Substituting these values in the formula for S.D. we have S.D. = `sqrt5/sqrt29 = sqrt(5/29)` units
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