NCERT Class 12 Mathematics Mathematics Part-II: Chapter 5 — THREE DIMENSIONAL GEOMETRY
This chapter introduces three-dimensional geometry using vector algebra, building upon concepts from Class XI Analytical Geometry and the previous chapter on vectors. It aims to simplify the study of 3D geometry through an elegant vector approach. The chapter covers direction cosines and direction ratios of a line, equations of lines and planes in space, angles between lines and planes, shortest distance between skew lines, and the distance of a point from a plane. Results are presented in vector form and translated into Cartesian form for clearer geometric and analytic understanding. Key concepts include direction angles, direction cosines (l, m, n), and direction ratios (a, b, c), along with their relationships and properties, aiding students in visualizing and solving problems in three-dimensional space for CBSE curriculum.
Quick info
| Board | CBSE / NCERT |
|---|---|
| Class | Class 12 |
| Subject | Mathematics |
| Book | Mathematics Part-II |
| Chapter | Chapter 5 — THREE DIMENSIONAL GEOMETRY |
| Language | English |
| PDF type | NCERT Textbook |
| Session | CBSE 2026 |
| Reading time | 3 minutes |
| Word count | 572 |
Learning outcomes
- Understand the concept of direction angles, direction cosines, and direction ratios of a line in 3D space.
- Relate direction cosines and direction ratios of a line.
- Derive and apply the equations of lines and planes in various forms.
- Calculate angles between lines, planes, and a line and a plane.
- Determine the shortest distance between two skew lines.
- Find the distance of a point from a plane.
Vocabulary
| Word | Meaning |
|---|---|
| Direction angles | The angles α, β, γ made by a directed line with the positive x, y, and z-axes, respectively. |
| Direction cosines | The cosines of the direction angles of a directed line, denoted by l, m, n. |
| Direction ratios | Any three numbers proportional to the direction cosines of a line, denoted by a, b, c. |
| Skew lines | Two lines in space that are neither parallel nor intersecting. |
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Practice questions
- If a line makes angles 90°, 135°, 45° with the x, y and z-axis respectively, find its direction cosines. Answer: The direction cosines are cos 90°, cos 135°, cos 45°, which are 0, -1/√2, 1/√2.
- Find the direction ratios of a line whose direction cosines are 1/√14, 2/√14, 3/√14. Answer: The direction ratios are proportional to the direction cosines. So, direction ratios can be 1, 2, 3 (by multiplying by √14).
- What are the direction cosines of a line parallel to the z-axis? Answer: A line parallel to the z-axis makes angles 90°, 90°, 0° with the axes. Its direction cosines are cos 90°, cos 90°, cos 0°, which are 0, 0, 1.
Practice MCQs
Q1. If a line has direction ratios -18, 12, -4, then its direction cosines are:
Explanation: First, find the magnitude of direction ratios: √((-18)² + 12² + (-4)²) = √(324 + 144 + 16) = √484 = 22. Then divide each ratio by 22: -18/22 = -9/11, 12/22 = 6/11, -4/22 = -2/11.
Q2. The direction cosines of a line are proportional to:
Explanation: Direction ratios are defined as any three numbers that are proportional to the direction cosines of a line.
Q3. If l, m, n are direction cosines of a line, then which of the following is true?
Explanation: The fundamental identity relating the direction cosines of any line in three-dimensional space is l² + m² + n² = 1.
Q4. What are the direction cosines of the x-axis?
Explanation: The x-axis makes an angle of 0° with itself and 90° with the y and z-axes. So, its direction cosines are cos 0°, cos 90°, cos 90°, which are 1, 0, 0.
Q5. If a, b, c are direction ratios of a line, then a/λ, b/λ, c/λ (where λ ≠ 0) represent:
Explanation: Any set of numbers proportional to the direction cosines are direction ratios. Dividing by a non-zero constant λ gives another set of direction ratios for the same line.
Frequently asked questions
What is the main purpose of using vector algebra in three-dimensional geometry as discussed in this chapter?
The purpose is to make the study of three-dimensional geometry simpler and more elegant by using vector algebra.
What are direction angles and direction cosines of a directed line?
Direction angles are the angles α, β, γ a directed line makes with the x, y, and z-axes, respectively. Their cosines (cos α, cos β, cos γ) are the direction cosines.
How are direction ratios related to direction cosines?
Any three numbers proportional to the direction cosines of a line are called its direction ratios. If l, m, n are direction cosines and a, b, c are direction ratios, then a = λl, b = λm, c = λn for some non-zero λ.
Can a line have multiple sets of direction cosines?
A given line in space can be extended in two opposite directions, leading to two sets of direction cosines. To have a unique set, the line must be considered as a directed line.
What is the fundamental relationship between the direction cosines of a line?
The sum of the squares of the direction cosines of any line is always equal to 1, i.e., l² + m² + n² = 1.
What are skew lines?
Skew lines are two lines in three-dimensional space that are neither parallel nor do they intersect each other.
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NCERT Class 12 Mathematics — Mathematics Part-II — Chapter 5 — THREE DIMENSIONAL GEOMETRY. Verified by NCERT Help Editorial Team. Reviewed on 29 Jul 2026. Last updated 10 Aug 2026.