---
title: "Symmetric form of line of two planes"
subject: "Mathematics"
chapter: 3
key: "semester_2/mathematics:symmetric-form-line-two-planes"
source: https://www.bctengineer.com/q/semester_2/mathematics/symmetric-form-line-two-planes
site: BCT Engineering
---

# Symmetric form of line of two planes

Mathematics · Chapter 3: Three Dimensional Solid Geometry

- Asked in: 2071 Bhadra, 5 marks
- Video: [EQUATION OF STRAIGHT LINE IN 3D || unsymmetrical/symmetrical/two-point form](https://www.youtube.com/watch?v=3UHefE46bNo) — Masoom Tutorials
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/symmetric-form-line-two-planes
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-3

## Question

Express the equation of the line x+2y+3z−6 = 0, 3x+4y+5z−2 = 0 in symmetric form.

## Short answer

Direction ratios from cross product of normals (1,2,3) and (3,4,5) are (1,-2,1). Setting z=0 gives point (-10,8,0). Symmetric form: (x+10)/1 = (y-8)/(-2) = z/1.

## Answer

**Given.** The line is the intersection of x + 2y + 3z − 6 = 0 and 3x + 4y + 5z − 2 = 0. To write it in symmetric form we need a point on it and its direction ratios.

#### Direction ratios
The line is perpendicular to both normals (1, 2, 3) and (3, 4, 5), so its direction is their cross product:

```text
d = (1, 2, 3) × (3, 4, 5)
  = (2·5 − 3·4, 3·3 − 1·5, 1·4 − 2·3)
  = (−2, 4, −2)  ∝ (1, −2, 1)
```

#### A point on the line
Put z = 0: x + 2y = 6 and 3x + 4y = 2. Subtract twice the first from the second: x = 2 − 12 = −10. Then y = 8. So (−10, 8, 0) lies on the line.

Check in the second plane: 3(−10) + 4(8) + 0 − 2 = 0 ✓.

#### Symmetric form
```text
(x + 10)/1 = (y − 8)/(−2) = z/1
```

**Answer: (x + 10)/1 = (y − 8)/(−2) = z/1.**

## Check yourself

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

1. What is needed for the symmetric form?
   A. A point on the line and its direction ratios
   B. Two points on each plane
   C. Only the normals of the planes
   D. The angle between the planes
   Answer: A. Symmetric form is (x - x0)/l = (y - y0)/m = (z - z0)/n. (Key point: Needed data)

2. How is the direction found?
   A. Dot product of the normals
   B. Cross product of the plane normals, (1, 2, 3) x (3, 4, 5) proportional to (1, -2, 1)
   C. Sum of the normals
   D. Difference of the plane constants
   Answer: B. The line is perpendicular to both normals. (Key point: Direction)

3. Putting z = 0, which point lies on the line?
   A. (-10, 8, 0)
   B. (10, -8, 0)
   C. (6, 0, 0)
   D. (-10, 8, 6)
   Answer: A. x + 2y = 6 and 3x + 4y = 2 give x = -10, y = 8. (Key point: A point)

4. What is the symmetric form?
   A. (x - 10)/1 = (y + 8)/(-2) = z/1
   B. (x + 10)/1 = (y - 8)/2 = z/1
   C. (x + 10)/1 = (y - 8)/(-2) = z/1
   D. (x + 10)/2 = (y - 8)/4 = z/6
   Answer: C. Using point (-10, 8, 0) and direction (1, -2, 1). (Key point: Result)

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