Q.259. Shortest between two skew lines in space.

Question 259.
Find out the shortest distance between the lines (x - 3)/3 = (y - 8)/1 = (z - 3)/-1 and
(x + 3)/3 = (y + 7)/-2 = (z - 6)/-4.

Answer 259.
The directions of the lines are (l, m, n) = (3, 1, -1) and (l', m', n') = (3, -2, -4).
As the directions are different, the lines are not parallel.
The two lines pass through a = (3, 8, 3) and b = (-3, -7, 6)
(a - b) . [(l, m, n) x (l', m', n')]
= [(3, 8, 3) - (-3, -7, 6)] . [(3, 1, -1) x (3, -2, -4)]
= (6, 15, -3) . (-6, 9, -9)
= -36 + 135 + 27
= 126 is not zero
=> the lines are skew.
Perpendicular distance between these skew lines
= l (a-b) . (l,m,n) x (l',m',n') l / l (l,m,n) x (l',m'n') l
= 126 / √(36 + 81 + 81)
= 8.95.

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