---
title: "Shortest distance is 1/√6"
subject: "Mathematics"
chapter: 3
key: "semester_2/mathematics:shortest-distance-1-sqrt-6"
source: https://www.bctengineer.com/q/semester_2/mathematics/shortest-distance-1-sqrt-6
site: BCT Engineering
---

# Shortest distance is 1/√6

Mathematics · Chapter 3: Three Dimensional Solid Geometry

- Asked in: 2069 Poush, 5 marks
- Video: [Shortest Distance Question | Skewline in shortest distance | Engineering Math | Prashant YT | PU |](https://www.youtube.com/watch?v=fFcqQs2tgho) — Prashant YT
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/shortest-distance-1-sqrt-6
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3

## Question

Show that the shortest distance between two skew lines (x−1)/2 = (y−2)/3 = (z−3)/4 and (x−2)/3 = (y−4)/4 = (z−5)/5 is 1/√6.

## Short answer

The shortest distance is |AB·(d1×d2)|/|d1×d2|. With points A(1,2,3), B(2,4,5) and directions (2,3,4) and (3,4,5), d1×d2=(-1,2,-1) with magnitude √6, and AB·(d1×d2)=1, so SD = 1/√6.

## Answer

For two skew lines r = a₁ + t d₁ and r = a₂ + s d₂, the shortest distance is the length of the projection of the join of two points on the common perpendicular:

SD = |(a₂ − a₁) · (d₁ × d₂)| / |d₁ × d₂|.

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

AB = (1, 2, 2).

#### Cross product
```text
d₁ × d₂ = (3·5 − 4·4, 4·3 − 2·5, 2·4 − 3·3)
        = (−1, 2, −1)
|d₁ × d₂| = √(1 + 4 + 1) = √6
```

#### Dot product
AB · (d₁ × d₂) = (1)(−1) + (2)(2) + (2)(−1) = −1 + 4 − 2 = 1.

#### Distance
SD = |1| / √6 = 1/√6.

Since d₁ × d₂ ≠ 0 the lines are not parallel, and since the scalar triple product is not zero they do not intersect, so they are skew.

**Hence the shortest distance is 1/√6.**

## Check yourself

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

1. The shortest distance between skew lines is:
   A. |d1 x d2| / |a2 - a1|
   B. |(a2 - a1) . (d1 x d2)| / |d1 x d2|
   C. |(a2 - a1) x (d1 x d2)|
   D. |d1 . d2| / |d1 x d2|
   Answer: B. It is the projection of the join vector on the common perpendicular direction. (Key point: Formula)

2. d1 = (2, 3, 4), d2 = (3, 4, 5): d1 x d2 =
   A. (-1, 2, -1)
   B. (1, 2, 1)
   C. (-1, -2, -1)
   D. (23, 26, 1)
   Answer: A. (15-16, 12-10, 8-9) = (-1, 2, -1), magnitude sqrt(6). (Key point: Cross product)

3. With AB = (1, 2, 2), AB . (d1 x d2) equals:
   A. 0
   B. -5
   C. 1
   D. 5
   Answer: C. -1 + 4 - 2 = 1. (Key point: Numerator)

4. Why are the lines skew?
   A. The triple product is zero
   B. d1 x d2 is nonzero and the triple product is nonzero
   C. d1 x d2 = 0
   D. d1 . d2 = 0
   Answer: B. Nonzero cross product means not parallel, nonzero triple product means no common plane. (Key point: Skewness)

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