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 is :

Solution

Maths sir · 4 steps

Cross the two directions to get the common normal, then project the joining vector onto it.

1. Read off the lines. Line 1 passes through A(3, 8, 3) with direction b⃗₁ = (3, -1, 1). Line 2 passes through B(-3, -7, 6) with direction b⃗₂ = (-3, 2, 4). Take care with signs: x + 3 means x = -3.
2. Common normal. b⃗₁ × b⃗₂ = ((-1)(4) - (1)(2), (1)(-3) - (3)(4), (3)(2) - (-1)(-3)) = (-6, -15, 3). Its length is √(36 + 225 + 9) = √270 = 3√30.

   d = (|AB⃗ · (b⃗₁ × b⃗₂)|)/(|b⃗₁ × b⃗₂|)

3. Put in numbers. AB⃗ = (-6, -15, 3), so the dot product is 36 + 225 + 9 = 270. Then d = 270/3√30 = 90/√30 = 3√30.
4. So. Here AB⃗ is exactly the common normal, so AB is itself the shortest segment, and |AB| = 3√30 confirms it. The answer is (c).

The idea to keep. For skew lines, the shortest distance is the projection of any joining vector onto b⃗₁ × b⃗₂. Get the points from the symmetric form with the correct signs before anything else.

Answer from the NTA final answer key. Solution by PrepWiser.