# NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry

## NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry

The concept of 3D geometry is very important for the students because it forms the base for many other higher level concepts. This chapter is important for both CBSE class 12 board exam as well as for the competitive exams like JEE Mains. Total of 36 questions in 3 exercises is given in this chapter. The NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry are designed and solved by maths experts to help the students to clear their doubts instantly.

In Class 11th, we already study Analytical Geometry in two dimensions, and only discuss the introduction of three-dimensional geometry, also in chapter 10 vector algebra of class 12, we have learnt some basic concepts of vectors. Now we will learn to use vector algebra in NCERT Class 12 Maths Chapter 11 Three Dimensional Geometry. The purpose of this approach to three-dimensional geometry is that it makes the study simple and effective. NCERT Chapter 11 Three-Dimensional geometry introduce a wide range of important topics like direction cosines and direction ratios of a line joining two points, equations of lines an Space, angle between two Lines, angle between two planes, angle between a line and a plane, shortest distance between two skew lines, equation of a plane in the normal form, etc.

 In this chapter we deal with formulas like- If l, m, n are the direction cosines of a line, then $\dpi{100} l^2+m^2+n^2=1$. $\dpi{100} Q(x_{2}, y_{2}, z_{2})$ and $\dpi{100} P(x_{1}, y_{1}, z_{1})$Direction cosines of a line joining two points $\dpi{80} PQ=\sqrt{(X_2-X_1)^2+(Y_2-Y_1)^2+(Z_2-Z_1)^2}$are $\dpi{80} \frac{X_2-X_1}{PQ},\:\frac{Y_2-Y_1}{PQ},\:\frac{Z_2-Z_1}{PQ}$, where If l, m, n are the direction cosines and a, b, c are the direction of a line then-              $\dpi{80} \dpi{80} l=\frac{a}{\sqrt{a^2+b^2+c^2}} ; m=\frac{b}{\sqrt{a^2+b^2+c^2}}; n=\frac{c}{\sqrt{a^2+b^2+c^2}}$

## Topics and sub-topics of NCERT Grade 12 Maths chapter-11 Three Dimensional Geometry

11.1 Introduction

11.2 Direction Cosines and Direction Ratios of a Line

11.2.1 Relation between the direction cosines of a line

11.2.2 Direction cosines of a line passing through two points

11.3 Equation of a Line in Space

11.3.1Equation of a line through a given point and parallel to a given vector b

11.3.2 Equation of a line passing through two given points

11.4 Angle between Two Lines

11.5 Shortest Distance between Two Lines

11.5.1 Distance between two skew lines

11.5.2 Distance between parallel lines

11.6 Plane

11.6.1 Equation of a plane in normal form

11.6.2 Equation of a plane perpendicular to a given vector and passing through a given point

11.6.3 Equation of a plane passing through three non collinear points

11.6.4 Intercept form of the equation of a plane

11.6.5 Plane passing through the intersection of two given planes

11.7 Coplanarity of Two Lines

11.8 Angle between Two Planes

11.9 Distance of a Point from a Plane

11.10 Angle between a Line and a Plane

## NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry- Exercise Questions

NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Exercise 11.1

NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Exercise 11.2

NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry Exercise 11.3

NCERT Solutions for class 12- Maths

 Chapter 1 Relations and Functions Chapter 2 Inverse Trigonometric Functions Chapter 3 Matrices Chapter 4 Determinants Chapter 5 Continuity and Differentiability Chapter 6 Application of Derivatives Chapter 7 Integrals Chapter 8 Application of Integrals Chapter 9 Differential Equations Chapter 10 Vector Algebra Chapter 12 Linear Programming Chapter 13 Probability