---
title: "Shortest distance between lines (6,7,4) and (0,−9,2)"
subject: "Mathematics"
chapter: 3
key: "semester_2/mathematics:shortest-distance-between-lines-6"
source: https://www.bctengineer.com/q/semester_2/mathematics/shortest-distance-between-lines-6
site: BCT Engineering
---

# Shortest distance between lines (6,7,4) and (0,−9,2)

Mathematics · Chapter 3: Three Dimensional Solid Geometry

- Asked in: 2073 Magh, 5 marks; 2069 Bhadra, 5 marks (asked 2 times)
- Video: [Find the length and equation of the shortest distance line between the given lines](https://www.youtube.com/watch?v=pP6Z-jWYEgQ) — Garg for Geeks
- 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-lines-6
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3

## Question

Find the S.D. between the lines (x−6)/3 = (y−7)/(−1) = (z−4)/1 and x/(−3) = (y+9)/2 = (z−2)/4. Find also the equation of shortest distance.

## Short answer

The shortest distance is 3√30 units, and the line of shortest distance is (x − 3)/2 = (y − 8)/5 = (z − 3)/(−1).

## Answer

**Lines.**
- L₁: (x − 6)/3 = (y − 7)/(−1) = (z − 4)/1, through A(6, 7, 4), direction d₁ = (3, −1, 1).
- L₂: x/(−3) = (y + 9)/2 = (z − 2)/4, through B(0, −9, 2), direction d₂ = (−3, 2, 4).

#### Common perpendicular direction
```text
n = d₁ × d₂ = (3, −1, 1) × (−3, 2, 4)
  = ((−1)(4) − (1)(2), (1)(−3) − (3)(4), (3)(2) − (−1)(−3))
  = (−6, −15, 3)
```

Since (−6, −15, 3) = −3(2, 5, −1), take n = (2, 5, −1), with |n| = √30.

#### Shortest distance
AB = (0 − 6, −9 − 7, 2 − 4) = (−6, −16, −2).

```text
S.D. = |AB · n| / |n|     with n = (2, 5, −1)
     = |−12 − 80 + 2| / √30
     = 90/√30
     = 3√30 ≈ 16.43
```

#### Equation of the shortest distance
Feet of the common perpendicular: A + s d₁ = (6 + 3s, 7 − s, 4 + s) on L₁ and B + u d₂ = (−3u, −9 + 2u, 2 + 4u) on L₂. Their difference must be perpendicular to both d₁ and d₂, which gives s = −1 and u = 1:

- foot on L₁: P(3, 8, 3)
- foot on L₂: Q(−3, −7, 6)

PQ = (−6, −15, 3) ∝ (2, 5, −1), and |PQ| = √(36 + 225 + 9) = √270 = 3√30 ✓.

The S.D. line is

```text
(x − 3)/2 = (y − 8)/5 = (z − 3)/(−1)
```

Equivalently, as two planes: the plane through L₁ and n is 4x − 5y − 17z + 79 = 0, and the plane through L₂ and n is 22x − 5y + 19z − 83 = 0.

**Answer: S.D. = 3√30 units; equation (x − 3)/2 = (y − 8)/5 = (z − 3)/(−1).**

## Check yourself

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

1. What is n = d1 x d2 (simplified)?
   A. (1, 1, 1)
   B. (2, 5, -1), up to a factor
   C. (2, -5, 1)
   D. (3, -1, 4)
   Answer: B. The cross product is (-6, -15, 3), proportional to (2, 5, -1). (Key point: Common perpendicular direction)

2. What is the S.D. using AB = (-6, -16, -2)?
   A. 90 sqrt(30)
   B. sqrt(30)
   C. 3 sqrt(30)
   D. 3 sqrt(10)
   Answer: C. |AB . n| = |-12 - 80 + 2| = 90, and 90/sqrt(30) = 3 sqrt(30). (Key point: S.D. value)

3. How are the feet found?
   A. Take general points on each line and require the difference to be perpendicular to d1 and d2
   B. Take the foot as the points A and B themselves
   C. Take the points where the lines cross the x-axis
   D. Take the midpoints of AB
   Answer: A. This gives s = -1 and u = 1, so P(3, 8, 3) and Q(-3, -7, 6). (Key point: Feet of the perpendicular)

4. What is the equation of the S.D. line?
   A. (x - 3)/2 = (y - 8)/5 = (z - 3)/(-1)
   B. (x - 3)/2 = (y - 8)/5 = (z - 3)/1
   C. (x - 6)/2 = (y - 7)/5 = (z - 4)/(-1)
   D. (x - 3)/3 = (y - 8)/(-1) = (z - 3)/1
   Answer: A. It passes through P(3, 8, 3) with direction (2, 5, -1). (Key point: Equation of S.D.)

## Close variants set in other exams

- [Shortest distance between lines (length and equation)](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Ashortest-distance-between-lines-length) — Mathematics, chapter 3, asked 7×
- [Shortest distance and its equation (lines 2,3,4 and 3,4,5)](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Ashortest-distance-its-equation-lines) — Mathematics, chapter 3, asked 7×

## Related in Mathematics

- [Shortest distance between two lines](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Ashortest-distance-between-two-lines) — Mathematics, chapter 3, asked 2×
- [Magnitude of shortest distance between lines](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Amagnitude-shortest-distance-between-lines) — Mathematics, chapter 3
- [Shortest distance is 1/√6](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Ashortest-distance-1-sqrt-6) — Mathematics, chapter 3, asked 2×
- [Magnitude and equation of S.D. between two lines](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Amagnitude-equation-s-d-between) — Mathematics, chapter 3, asked 2×