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The equation of the line of shortest distance between the lines L1: (x-6)/3 = (y-7)/(-1) = (z-4)/1 and L2: x/(-3) = (y+9)/2 = (z-2)/4 is:

1. (x-6)/2 = (y-7)/5 = (z-4)/(-1)
2. x/2 = (y+9)/5 = (z-2)/(-1)
3. (x+3)/2 = (y+7)/5 = (z+6)/(-1)
4. (x-3)/2 = (y-8)/5 = (z-3)/(-1)

Correct answer: (x-6)/2 = (y-7)/5 = (z-4)/(-1)

Solution

Direction vectors: d1=(3,-1,1), d2=(-3,2,4). SD direction = d1 x d2 = |i j k; 3 -1 1; -3 2 4| = i*(-4-2) - j*(12+3) + k*(6-3) = (-6,-15,3), proportional to (2,5,-1). The point (6,7,4) lies on L1 (it is the given point at parameter 0). Option A: (x-6)/2=(y-7)/5=(z-4)/(-1) passes through (6,7,4) with direction (2,5,-1), which matches. This is the SD line at the foot on L1.

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