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Find the shortest distance between the two lines L1: (x-3)/4 = (y+7)/(-11) = (z-1)/5 and L2: (x-5)/3 = (y-9)/(-6) = (z+2)/1.

1. 187/sqrt(563)
2. 178/sqrt(563)
3. 185/sqrt(563)
4. 179/sqrt(563)

Correct answer: 178/sqrt(563)

Solution

Direction vectors: d1 = (4,-11,5), d2 = (3,-6,1). d1 x d2 = |i j k; 4 -11 5; 3 -6 1| = i((-11)(1)-(5)(-6)) - j((4)(1)-(5)(3)) + k((4)(-6)-(-11)(3)) = i(-11+30) - j(4-15) + k(-24+33) = 19i + 11j + 9k. |d1 x d2| = sqrt(361+121+81) = sqrt(563). Vector r2 - r1 = (5-3, 9-(-7), -2-1) = (2, 16, -3). (r2-r1).(d1 x d2) = 2*19 + 16*11 + (-3)*9 = 38 + 176 - 27 = 187. Shortest distance = 187/sqrt(563).

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