 # NCERT Solution Class 12 Chapter 11- Three Dimensional Geometry

## NCERT Solutions For Class 12 Maths Chapter 11 PDF Free Download

The NCERT Solutions for Class 12 Maths Chapter 11- Three Dimensional Geometry are available here to help the students understand the ways in which different questions should be solved. Looking around, we can observe that everything in the real world is in a three-dimensional shape. In this chapter students learn about three dimensional geometry in detail. Students can easily score full marks in the questions from this chapter by solving all the questions present in the NCERT textbook.

The class 12 NCERT solutions for Three Dimensional Geometry are very easy to understand. These solutions cover all the exercise questions included in the book and are according to the latest guidelines of CBSE. Here, the PDF of the class 12 maths chapter 11 NCERT solutions is available which can be downloaded and referred in online mode.

### Access Answers of Maths NCERT Class 12 Chapter 11- Three Dimensional Geometry                                            Also Access NCERT Exemplar for Class 12 Maths Chapter 11 CBSE Notes for Class 12 Maths Chapter 11

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

The major concepts of Maths covered in Chapter 11- Three Dimensional Geometry of NCERT Solutions for Class 12 includes:

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

11.3.1 Equation 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, 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

Exercise 11.1 Solutions 5 Questions

Exercise 11.2 Solutions 17 Questions

Exercise 11.3 Solutions 14 Questions

Miscellaneous Exercise On Chapter 11 Solutions 23 Questions

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

The chapter Three Dimensional Geometry belongs to the unit Vectors and Three – Dimensional Geometry, that adds up to 14 marks of the total marks. There are 3 exercises along with a miscellaneous exercise in this chapter to help students understand the concepts related to Three Dimensional Geometry clearly. Some of the topics discussed in Chapter 11 of NCERT Solutions for Class 12 Maths are as follows:

1. Direction cosines of a line are the cosines of the angles made by the line with the positive directions of the coordinate axes.
2. If l, m, n are the direction cosines of a line, then l2 + m2 + n2 = 1
3. Direction ratios of a line are the numbers which are proportional to the direction cosines of a line.
4. Skew lines are lines in space which are neither parallel nor intersecting. They lie in different planes.
5. Angle between skew lines is the angle between two intersecting lines drawn from any point (preferably through the origin) parallel to each of the skew lines.
6. If l1 , m1, n1 and l2, m2, n2 are the direction cosines of two lines; and θ is the acute angle between the two lines; then cosθ = |l1l2 + m1m2 + n1n2|

These are very few topics that are discussed in the chapter three dimensional geometry. To know more about the chapter, refer the NCERT Textbook of Class 12 Maths.

### Key Features of NCERT Solutions for Class 12 Maths Chapter 11- Three Dimensional Geometry

Studying the Three Dimensional Geometry of Class 12 enables the students to understand the following:

Direction cosines and direction ratios of a line joining two points.Cartesian equation and vector equation of a line, coplanar and skew lines, shortest distance between two lines.Cartesian and vector equation of a plane. Angle between (i) two lines, (ii) two planes, (iii) a line and a plane.Distance of a point from a plane.