Find the shortest distance between the skew lines L_1: (x-2)/1=(y+3)/2=(z-1)/-1 and L_2: (x+1)/3=(y-4)/-1=(z+2)/2.

Find the shortest distance between the skew lines $L_{1} : \frac{x - 2}{1} = \frac{y + 3}{2} = \frac{z - 1}{- 1}$ and $L_{2} : \frac{x + 1}{3} = \frac{y - 4}{- 1} = \frac{z + 2}{2}$ .

hard 4 marks Class 12 · Mathematics · Three Dimensional Geometry

Answer

23/√83

The correct answer is 23/√83 — Thus the distance between the skew lines is $23/ 83$ .

Write parametric equations: $L_{1} : (x, y, z) = (2, - 3, 1) + t (1, 2, - 1)$ and $L_{2} : (x, y, z) = (- 1, 4, - 2) + s (3, - 1, 2)$ .
Determine direction vectors: $d_{1} = (1, 2, - 1)$ , $d_{2} = (3, - 1, 2)$ .
Compute cross product $d_{1} \times d_{2} = (3, - 5, - 7)$ .
Choose a point on each line: $P_{1} = (2, - 3, 1)$ , $P_{2} = (- 1, 4, - 2)$ .
Form vector $w = P_{2} - P_{1} = (- 3, 7, - 3)$ .
Calculate the scalar triple product: $∣ w c d o t (d_{1} \times d_{2}) ∣ = ∣ (- 3) (3) + 7 (- 5) + (- 3) (- 7) ∣ = ∣ - 9 - 35 + 21∣ = 23$ .
Find the magnitude of the cross product: $∣ d_{1} \times d_{2} ∣ = 3^{2} + (- 5)^{2} + (- 7)^{2} = 83$ .
The shortest distance equals $23/ 83$ .
Thus the distance between the skew lines is $23/ 83$ .

Common mistake: Forgetting the absolute value of the scalar triple product or using the wrong cross‑product orientation

Concept tested: Shortest distance between skew lines

Published 10 September 2026 · Last updated 16 September 2026 · How this solution was prepared and checked