The shortest distance between the lines (x - 1)/2 = (y + 8)/-7 = (z - 4)/5 and (x - 1)/2 = (y - 2)/1 = (z - 6)/(- 3) is :

Solution

Maths sir · 3 steps

Join a point on each line, then project that join onto the common perpendicular b⃗₁ × b⃗₂.

1. Read off the lines. Line 1 passes through A(1, -8, 4) with direction b⃗₁ = (2, -7, 5). Line 2 passes through B(1, 2, 6) with direction b⃗₂ = (2, 1, -3). So AB⃗ = (0, 10, 2).
2. Common perpendicular. b⃗₁ × b⃗₂ = ((-7)(-3) - 5(1), 5(2) - 2(-3), 2(1) - (-7)(2)) = (16, 16, 16). This is 16(1, 1, 1), so its magnitude is 16√3.

   d = (|AB⃗ · (b⃗₁ × b⃗₂)|)/(|b⃗₁ × b⃗₂|) = (|0 + 160 + 32|)/16√3 = 192/16√3 = 12/√3

3. So. d = 12/√3 = 4√3, which is option (c).

The idea to keep. For skew lines, the shortest distance is the projection of any joining vector onto b⃗₁ × b⃗₂. If the cross product has a common factor, pull it out first: the arithmetic gets much easier.

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