---
title: "Shortest distance between two lines"
subject: "Mathematics"
chapter: 3
key: "semester_2/mathematics:shortest-distance-between-two-lines"
source: https://www.bctengineer.com/q/semester_2/mathematics/shortest-distance-between-two-lines
site: BCT Engineering
---

# Shortest distance between two lines

Mathematics · Chapter 3: Three Dimensional Solid Geometry

- Asked in: 2075 Baisakh, 5 marks
- Video: [Find the shortest distance between the lines: (x + 1)/2 = (y - 1)/1 = (z - 9)/- 3 and (x - 3)/2 = (y](https://www.youtube.com/watch?v=84HrN5WvKPc) — Mathematics Untold
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/shortest-distance-between-two-lines
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3

## Question

Show that the shortest distance between the lines x+a = 2y = −12z and x = y+2a = 6z−6a is 2a.

## Short answer

Take points A(-a,0,0) on L1, B(0,-2a,a) on L2. Direction vectors d1=(12,6,-1), d2=(6,6,1). n = d1×d2 = (12,-18,36) ∝ (2,-3,6). Shortest distance = |AB·(2,-3,6)|/7 = |2a+6a+6a|/7 = 2a.

## Answer

**Lines.**
- L₁: x + a = 2y = −12z. Divide by 12: (x + a)/12 = y/6 = z/(−1). Through A(−a, 0, 0), direction d₁ = (12, 6, −1).
- L₂: x = y + 2a = 6z − 6a. Divide by 6: x/6 = (y + 2a)/6 = (z − a)/1. Through B(0, −2a, a), direction d₂ = (6, 6, 1).

#### Common perpendicular direction
```text
n = d₁ × d₂ = (6·1 − (−1)·6, (−1)·6 − 12·1, 12·6 − 6·6)
            = (12, −18, 36)  ∝ (2, −3, 6)
|n| = 7 (for (2, −3, 6));  using (12, −18, 36): |n| = 6 × 7 = 42
```

#### Shortest distance
AB = B − A = (a, −2a, a).

```text
S.D. = |AB · n| / |n|
     = |a(2) + (−2a)(−3) + a(6)| / 7      using n = (2, −3, 6)
     = |2a + 6a + 6a| / 7
     = 14a / 7
     = 2a
```

Hence the shortest distance between the lines is 2a. ∎

(Equivalently with n = (12, −18, 36): |12a + 36a + 36a|/42 = 84a/42 = 2a.)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What are the direction ratios of L1: x + a = 2y = -12z?
   A. (1, 2, -12)
   B. (12, 6, 1)
   C. (12, 6, -1)
   D. (1, 1/2, -1/12)
   Answer: C. Dividing by 12 gives (x + a)/12 = y/6 = z/(-1). (Key point: Symmetric form)

2. What is d1 x d2 for d1 = (12, 6, -1), d2 = (6, 6, 1)?
   A. (12, -18, 36), proportional to (2, -3, 6)
   B. (12, 18, 36)
   C. (72, 6, -1)
   D. (6, 6, 36)
   Answer: A. The cross product components are 6 + 6, -6 - 12, and 72 - 36. (Key point: Common perpendicular)

3. What is AB for A(-a, 0, 0) and B(0, -2a, a)?
   A. (-a, 2a, -a)
   B. (a, -2a, a)
   C. (a, 2a, a)
   D. (a, -2a, 0)
   Answer: B. B - A = (a, -2a, a). (Key point: Vector between points)

4. What does |AB . n|/|n| give with n = (2, -3, 6)?
   A. 14a
   B. a
   C. 2a/7
   D. 2a
   Answer: D. |2a + 6a + 6a|/7 = 14a/7 = 2a. (Key point: Result)

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