---
title: "Shortest distance and its equation (lines 2,3,4 and 3,4,5)"
subject: "Mathematics"
chapter: 3
key: "semester_2/mathematics:shortest-distance-its-equation-lines"
source: https://www.bctengineer.com/q/semester_2/mathematics/shortest-distance-its-equation-lines
site: BCT Engineering
---

# Shortest distance and its equation (lines 2,3,4 and 3,4,5)

Mathematics · Chapter 3: Three Dimensional Solid Geometry

- Asked in: 2068 Magh, 5 marks
- 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-its-equation-lines
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3

## Question

Find the shortest distance between the lines (x−3)/2 = (y−4)/3 = (z−5)/4 and (x−4)/3 = (y−5)/4 = (z−7)/5. Also find the equation of the shortest distance line.

## Short answer

The shortest distance is 1/√6 units. The line of shortest distance is the intersection of the planes 11x + 2y - 7z - 6 = 0 and 7x + y - 5z + 2 = 0.

## Answer

Line 1: point A(3, 4, 5), direction d₁ = (2, 3, 4).
Line 2: point B(4, 5, 7), direction d₂ = (3, 4, 5).

#### Shortest distance
```text
d₁ × d₂ = (3·5 − 4·4, 4·3 − 2·5, 2·4 − 3·3) = (−1, 2, −1)
|d₁ × d₂| = √6
AB = (1, 1, 2)
AB · (d₁ × d₂) = −1 + 2 − 2 = −1
SD = |−1| / √6 = 1/√6
```

#### Equation of the line of shortest distance
It is the intersection of two planes:
- Plane 1 contains line 1 and the common perpendicular direction n = (−1, 2, −1).
- Plane 2 contains line 2 and the same direction n.

Normal of plane 1 = d₁ × n:

```text
(2,3,4) × (−1,2,−1) = (3(−1) − 4(2), 4(−1) − 2(−1), 2·2 − 3(−1))
                    = (−11, −2, 7)
```
Plane 1 through A(3, 4, 5): −11(x − 3) − 2(y − 4) + 7(z − 5) = 0, i.e. **11x + 2y − 7z − 6 = 0**.

Normal of plane 2 = d₂ × n:

```text
(3,4,5) × (−1,2,−1) = (4(−1) − 5(2), 5(−1) − 3(−1), 3·2 − 4(−1))
                    = (−14, −2, 10)  ∥  (7, 1, −5)
```
Plane 2 through B(4, 5, 7): 7(x − 4) + (y − 5) − 5(z − 7) = 0, i.e. **7x + y − 5z + 2 = 0**.

Check: A satisfies plane 1: 33 + 8 − 35 − 6 = 0; B satisfies plane 2: 28 + 5 − 35 + 2 = 0.

**Answer: SD = 1/√6, and the line of shortest distance is 11x + 2y − 7z − 6 = 0 = 7x + y − 5z + 2.**

## Check yourself

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

1. The shortest distance between the two lines is:
   A. 1/sqrt(6)
   B. 1/6
   C. sqrt(6)
   D. 1
   Answer: A. |AB . (d1 x d2)| / |d1 x d2| = 1/sqrt(6). (Key point: SD value)

2. How is its equation obtained?
   A. As the line joining A and B
   B. As the intersection of the plane containing line 1 and n, and the plane containing line 2 and n
   C. As the line through the origin along n
   D. As the average of the two lines
   Answer: B. n = d1 x d2 is the common perpendicular direction. (Key point: Line of shortest distance)

3. The normal of plane 1 is:
   A. d1 . n
   B. d1 + n
   C. d1 x n = (-11, -2, 7)
   D. n x n
   Answer: C. The plane contains d1 and n, so the normal is their cross product. (Key point: Plane 1 normal)

4. The line of shortest distance is:
   A. 11x + 2y - 7z - 6 = 0 = 7x + y - 5z + 2
   B. 11x + 2y - 7z + 6 = 0 = 7x + y - 5z - 2
   C. 7x + y - 5z + 2 = 0 only
   D. 11x - 2y + 7z - 6 = 0 = 7x + y - 5z + 2
   Answer: A. Plane 1 through A(3, 4, 5) and plane 2 through B(4, 5, 7). (Key point: Result)

## 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 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×

## Related in Mathematics

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