---
title: "Condition for lines to be perpendicular"
subject: "Mathematics"
chapter: 3
key: "semester_2/mathematics:condition-lines-be-perpendicular"
source: https://www.bctengineer.com/q/semester_2/mathematics/condition-lines-be-perpendicular
site: BCT Engineering
---

# Condition for lines to be perpendicular

Mathematics · Chapter 3: Three Dimensional Solid Geometry

- Asked in: 2067 Chaitra, 5 marks
- Video: [Parallel and perpendicular two lines in 3D | Angle Between Two Lines | Easy Method](https://www.youtube.com/watch?v=f4DafgIkcP0) — Maths With Hanif
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/condition-lines-be-perpendicular
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3

## Question

Prove that the lines x = ay+b, z = cy+d and x = a'y+b', z = c'y+d' are perpendicular if aa' + cc' + 1 = 0.

## Short answer

Rewrite each line in symmetric form to obtain direction ratios (a,1,c) and (a',1,c'). The dot product a·a' + 1·1 + c·c' = 0 simplifies to aa' + cc' + 1 = 0, which is the perpendicularity condition.

## Answer

Two lines are perpendicular if the dot product of their direction ratios is zero.

#### Direction ratios
The line x = ay + b, z = cy + d can be written in symmetric form by solving for y:

(x − b)/a = y/1 = (z − d)/c.

So its direction ratios are (a, 1, c). (Equivalently, each coordinate is a function of y: x = ay + b, z = cy + d, so for a change of 1 in y, x changes by a and z by c.)

Similarly the second line x = a′y + b′, z = c′y + d′ has direction ratios (a′, 1, c′).

#### Condition
The lines are perpendicular when

```text
a·a′ + 1·1 + c·c′ = 0
```

**Hence the lines are perpendicular if aa′ + cc′ + 1 = 0.**

Example: x = 2y, z = y and x = −y, z = −y: aa′ + cc′ + 1 = −2 − 1 + 1 = −2 ≠ 0, not perpendicular; but x = y, z = y and x = −y, z = 0·y gives (1)(−1) + (1)(0) + 1 = 0, perpendicular.

## Check yourself

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

1. Two lines are perpendicular when:
   A. Their direction ratios are equal
   B. The dot product of their direction ratios is zero
   C. Their cross product is zero
   D. Their dot product is 1
   Answer: B. Perpendicular directions have zero dot product. (Key point: Perpendicular criterion)

2. Its direction ratios are:
   A. (1, a, c)
   B. (a, c, 1)
   C. (a, 1, c)
   D. (b, 1, d)
   Answer: C. Per unit change of y, x changes by a and z by c. (Key point: Direction of x = ay+b, z = cy+d)

3. The dot product (a, 1, c) . (a', 1, c') = 0 gives:
   A. aa' + cc' + 1 = 0
   B. aa' + cc' = 0
   C. aa' + cc' - 1 = 0
   D. a + c + 1 = 0
   Answer: A. aa' + 1 + cc' = 0. (Key point: Resulting condition)

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