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

# Shortest distance between lines (length and equation)

Mathematics · Chapter 3: Three Dimensional Solid Geometry

- Asked in: 2074 Bhadra, 5 marks; 2073 Bhadra, 5 marks; 2070 Bhadra, 5 marks; 2067 Mangsir, 5 marks (asked 4 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-length
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3

## Question

Find the length and equation of the shortest distance between the lines (x−3)/3 = (y−8)/(−1) = (z−3)/1 and 2x−3y+27 = 0, 2y−z+20 = 0.

## Answer

#### Put both lines in symmetric form
**Line 1:** (x − 3)/3 = (y − 8)/(−1) = (z − 3)/1. Point A(3, 8, 3), direction d₁ = (3, −1, 1).

**Line 2:** 2x − 3y + 27 = 0, 2y − z + 20 = 0. Put y = 9: x = 0, z = 38, so B(0, 9, 38) is on it. Its direction is perpendicular to both normals (2, −3, 0) and (0, 2, −1):

d₂ = (2, −3, 0) × (0, 2, −1) = (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, −9, 9)
```
So take n ∝ (2, 3, −3), with |n| = √(4 + 9 + 9) = √22.

#### Length of S.D.
AB = (0 − 3, 9 − 8, 38 − 3) = (−3, 1, 35).

```text
S.D. = |AB · n| / |n|
     = |(−3)(2) + (1)(3) + (35)(−3)| / √22
     = |−6 + 3 − 105| / √22
     = 108/√22
     = 54√22/11 ≈ 23.03
```

#### Equation of the shortest distance
It is the intersection of two planes, each containing one line and the direction n = (2, 3, −3).

- Plane through line 1 (point A, directions d₁ and n): normal = d₁ × n = (3, −1, 1) × (2, 3, −3) = (0, 11, 11) ∝ (0, 1, 1). Plane: (y − 8) + (z − 3) = 0, i.e. **y + z = 11**.
- Plane through line 2 (point B, directions d₂ and n): normal = d₂ × n = (3, 2, 4) × (2, 3, −3) = (−18, 17, 5). Plane: −18x + 17(y − 9) + 5(z − 38) = 0, i.e. **18x − 17y − 5z + 343 = 0**.

So the shortest distance line is

```text
y + z = 11,   18x − 17y − 5z + 343 = 0
```

Its feet are (−90/11, 129/11, −8/11) on line 1 and (−18, −3, 14) on line 2, so in symmetric form: (x + 18)/2 = (y + 3)/3 = (z − 14)/(−3).

**Answer: S.D. = 54√22/11 ≈ 23.03 units.**

## Check yourself

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

1. How is the direction of line 2 (given as two planes) found?
   A. Add the normals
   B. Cross product of the two plane normals, giving (3, 2, 4)
   C. Take the normal of the first plane
   D. Cross product of the two lines' points
   Answer: B. The line lies in both planes so it is perpendicular to both normals (2,-3,0) and (0,2,-1). (Key point: Direction of line 2)

2. What gives the direction of the shortest distance?
   A. d1 x d2, proportional to (2, 3, -3)
   B. d1 . d2
   C. d1 + d2
   D. AB, the vector between the points
   Answer: A. The shortest distance is perpendicular to both lines, so along their cross product. (Key point: Common perpendicular direction)

3. How is the length found, and what is it?
   A. |AB x n| = 54 sqrt(22)/11
   B. |AB . n| = 108
   C. |AB . n|/|n| = 54 sqrt(22)/11
   D. |AB|/|n| = 54 sqrt(22)/11
   Answer: C. The projection of AB on the unit common perpendicular gives 108/sqrt(22). (Key point: S.D. formula and value)

4. How is the equation of the shortest distance line obtained?
   A. As the line through the two given points A and B
   B. As the line through A with direction d1
   C. As the line through B with direction n only
   D. As the intersection of two planes, each containing one line and the direction n
   Answer: D. Each plane contains one of the lines and the common perpendicular direction, so their intersection is the S.D. line. (Key point: Equation of S.D.)

## Close variants set in other exams

- [Shortest distance between lines (6,7,4) and (0,−9,2)](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Ashortest-distance-between-lines-6) — 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

- [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 between two lines](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Ashortest-distance-between-two-lines) — Mathematics, chapter 3, asked 2×
- [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×
- [Convergence of 2x+3x²/8+4x³/27+...](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aconvergence-2x-3x2-8-4x3) — Mathematics, chapter 6, asked 3×
- [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×
- [Coplanar lines and point of intersection](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Acoplanar-lines-point-intersection) — Mathematics, chapter 3, asked 2×

## Related in other subjects

- [Solve y''+3y'+2y=e^(2x) sin x](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Ade-d2-3d-2-e2x-sinx) — Mathematics I, chapter 4, asked 2×
- [Equation and length of axes of x²−4xy−2y²+10x+4y=0](https://www.bctengineer.com/question.md?key=semester_1%2Fmathematics_i%3Aconic-x2-4xy-2y2-equation-axes) — Mathematics I, chapter 3
- [Direct form I and II of 3y[n]+y[n−1]+2y[n−4]](https://www.bctengineer.com/question.md?key=semester_7%2Fdigital_signal_analysis_and_processing%3Adirect-form-1-2-delay4) — Digital Signal Analysis and Processing, chapter 4, asked 4×
- [Solve ODE by Laplace transform y''+3y'+2y=eᵗ](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Asolve-ode-by-laplace-transform-y-three) — Mathematics III, chapter 3, asked 4×