The shortest distance between the lines
(x - 5)/1 = (y - 2)/2 = (z - 4)/(- 3) and
(x + 3)/1 = (y + 5)/4 = (z - 1)/(- 5) is :

Solution

Maths sir · 4 steps

Join one point from each line, then project that joining vector onto d⃗₁ × d⃗₂.

1. Read off the lines. Line 1 passes through A(5, 2, 4) with direction d⃗₁ = (1, 2, -3). Line 2 passes through B(-3, -5, 1) with direction d⃗₂ = (1, 4, -5).

   d = (|AB⃗ · (d⃗₁ × d⃗₂)|)/(|d⃗₁ × d⃗₂|)

2. Cross product. d⃗₁ × d⃗₂ = ((2)(-5) - (-3)(4), (-3)(1) - (1)(-5), (1)(4) - (2)(1)) = (2, 2, 2). Its length is √(4 + 4 + 4) = 2√3.
3. Dot product. AB⃗ = (-3 - 5, -5 - 2, 1 - 4) = (-8, -7, -3). Then AB⃗ · (2, 2, 2) = -16 - 14 - 6 = -36.
4. So. d = 36/2√3 = 18/√3 = 6√3, which is option (d).

The idea to keep. The cross product of the two directions is perpendicular to both lines. The shortest distance is the length of the projection of any joining vector onto that perpendicular.

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