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# Three Dimensional Geometry

Three Dimensional Geometry is chapter 5 in Mathematics Part - II, a CBSE Class 12 book for the subject Mathematics on Edzy.

## Knowledge Snapshot

| Field | Details |
| :--- | :--- |
| Board | CBSE |
| Curriculum | CBSE |
| Class | Class 12 |
| Subject | Mathematics |
| Book | Mathematics Part - II |
| Chapter | Three Dimensional Geometry |
| Chapter Number In Book | 5 |
| Chapter Slug | three-dimensional-geometry |
| Pages | 377-395 |
| Page Count | 19 |
| Source PDF | https://edzy-ai.s3.ap-south-1.amazonaws.com/edzy-express-ts/d690ba5c-014d-4de2-90ba-7c55f5a5f0bf.pdf |
| Last Updated | 2026-06-25 |

## Chapter Overview

In this chapter, we shall study the direction cosines and direction ratios of a line joining two points, the equations of lines and planes in space under different conditions, the angle between two lines and two planes, the angle between a line and a plane, the shortest distance between two skew lines, and the distance of a point from a plane. Most of the results are obtained in vector form, though Cartesian form will also be utilized for clarity.

## Topic Inventory

| No. | Topic ID | Topic |
| :--- | :--- | :--- |
| 1 | 69aa6091a243cecf961fb724 | Introduction |
| 2 | 69aa6091a243cecf961fb728 | Direction Cosines and Direction Ratios of a Line |
| 3 | 69aa6091a243cecf961fb72b | Equation of a Line in Space |
| 4 | 69aa6091a243cecf961fb730 | Angle between Two Lines |
| 5 | 69aa6091a243cecf961fb736 | Shortest Distance between Two Lines |

## Topic-Wise Extracted Content

### Introduction
In Class XI, we studied Analytical Geometry in two dimensions and began with an introduction to three-dimensional geometry, focusing on Cartesian methods. We will now apply vector algebra to simplify the study of three-dimensional geometry.

### Direction Cosines and Direction Ratios of a Line
Direction cosines are defined as cosines of the angles that a directed line makes with the x, y, and z-axes, known as direction angles. When the direction of the line is reversed, the direction angles are replaced by their supplements, altering the signs of the direction cosines. A given line has two sets of direction cosines, and unique direction cosines for a line are denoted by l, m, and n. The three numbers proportional to the direction cosines are called direction ratios. If a, b, c are direction ratios, the direction cosines can be expressed as \( l = \pmrac{a}{\sqrt{a^2 + b^2 + c^2}}, m = \pmrac{b}{\sqrt{a^2 + b^2 + c^2}}, n = \pmrac{c}{\sqrt{a^2 + b^2 + c^2}} \). Direction ratios can also be scaled, producing an infinite number of proportional sets.

### Equation of a Line in Space
A line in space can be represented in vector form as \( \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \), where \( \mathbf{a} \) is the position vector of a point on the line and \( \mathbf{b} \) is a direction vector. Eliminating the parameter \( \lambda \), we arrive at the Cartesian equation \( rac{x - x_1}{a} = rac{y - y_1}{b} = rac{z - z_1}{c} \), where \( a, b, c \) are direction ratios of the line.

### Angle between Two Lines
This topic is introduced but requires further development in context.

### Shortest Distance between Two Lines
This topic is introduced but requires further development in context.

## Chapter Summary

The chapter covers the fundamental aspects of three-dimensional geometry with a focus on vector algebra. Key topics include direction cosines, direction ratios, equations of lines, and the shortest distance between lines, thereby facilitating a comprehensive understanding of spatial relationships in geometry.

## Key Ideas

- Direction cosines correspond to angles made with coordinate axes.
- Direction ratios are proportional to direction cosines.
- Lines in space can be described using vector or Cartesian equations.

## Important Chapter Facts

- The chapter is Three Dimensional Geometry.
- The chapter number in the book is 5.
- The subject is Mathematics.
- The book is Mathematics Part - II.
- The class is Class 12.
- The source PDF is https://edzy-ai.s3.ap-south-1.amazonaws.com/edzy-express-ts/d690ba5c-014d-4de2-90ba-7c55f5a5f0bf.pdf.
- The chapter covers pages 377-395.
- The chapter page count is 19.

## Related OKF Entities

| Entity Type | Title | URL |
| :--- | :--- | :--- |
| Chapter Canonical Page | Three Dimensional Geometry | https://www.edzy.ai/cbse-class-12-mathematics-mathematics-part-ii-three-dimensional-geometry |
| Chapter Markdown | Three Dimensional Geometry | https://www.edzy.ai/okf/chapter/cbse-class-12-mathematics-mathematics-part-ii-three-dimensional-geometry.md |
| Source PDF | Three Dimensional Geometry | https://edzy-ai.s3.ap-south-1.amazonaws.com/edzy-express-ts/d690ba5c-014d-4de2-90ba-7c55f5a5f0bf.pdf |

## Source Attribution

| Field | Value |
| :--- | :--- |
| Source | Edzy |
| Reference Type | examSubjectBookChapter |
| Reference ID | 66dfdfbd3f8b4e9e69bf7f94 |
| Chapter ID | 66dfdfbd3f8b4e9e69bf7f94 |
| Canonical URL | https://www.edzy.ai/cbse-class-12-mathematics-mathematics-part-ii-three-dimensional-geometry |
| Markdown URL | https://www.edzy.ai/okf/chapter/cbse-class-12-mathematics-mathematics-part-ii-three-dimensional-geometry.md |
| Source PDF | https://edzy-ai.s3.ap-south-1.amazonaws.com/edzy-express-ts/d690ba5c-014d-4de2-90ba-7c55f5a5f0bf.pdf |
| Last Updated | 2026-06-25 |