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The equation of the line passing through (−4, 3, 1), parallel to the plane x + 2y − z − 5 = 0 and intersecting the line (x+1)/(−3) = (y−3)/2 = (z−2)/(−1) is:

1. (x−4)/2 = (y+3)/1 = (z+1)/4
2. (x+4)/1 = (y−3)/1 = (z−1)/3
3. (x+4)/3 = (y−3)/(−1) = (z−1)/1
4. (x+4)/(−1) = (y−3)/1 = (z−1)/1

Correct answer: (x+4)/3 = (y−3)/(−1) = (z−1)/1

Solution

Required line passes (-4,3,1) and must be parallel to x+2y-z=0 (direction.normal = 0) and intersect the line (x+1)/-3=(y-3)/2=(z-2)/-1. Testing the choices, direction (3,-1,1) satisfies (3)(1)+(-1)(2)+(1)(-1)=0 and meets the given line at (2,1,3). Hence the line is (x+4)/3 = (y-3)/(-1) = (z-1)/1.

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