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# NCERT Solutions for Class 12 Maths Chapter 11 - PDF Download `

NCERT solution for class 12 mathematics chapter 11 3 Dimensional Geometry describes vector algebra of 3D geometry, Which is really important for students who want to go into science and engineering. Chapter 11 of NCERT class 12 maths concentrates on Three-dimensional Geometry, which provides students with a comprehensive introduction to the concept of three-dimensional space and the relationships between coordinates, vectors and planes. This chapter serves as a basis for the comprehension of complex concepts. NCERT solutions for class 12 maths chapter 11 explain the exercises in detail. NCERT solutions are created by qualified mathematicians and teachers of eSaral to make sure the concepts are clear and correct.

The NCERT class 12 mathematics chapter-11 PDF is available for free to download from eSaral’s official website. This PDF provides answers to all the questions, problems and solutions related to class 12 mathematics chapter 11. Similarly, the Three-dimensional Geometry class 12 PDF provides all the answers related to 3D Geometry. Our NCERT Solutions for 3D Geometry Class 12 are written by subject experts with years of teaching experience, so you can trust that they'll make sure you understand the questions perfectly by giving you all the justifications and solutions you need.

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

In mathematics, three-dimensional geometry refers to the study of the properties of a line, a point, and different solid shapes in a system of three-dimensional coordinates. NCERT solutions for chapter 11 of class 12 maths are designed to help you to understand these concepts and how to use them to answer different kinds of questions.

### Important Topics Covered in Chapter 11 - Three Dimensional Geometry

NCERT solutions Class 12 Chapter 10 has some really important topics to cover.

11.1 Introduction - Mathematics class 12 chapter 11 will provide an exploration of Analytical Geometrical Analysis in two dimensions, as well as the use of Cartesian Methods in three dimensions. The chapter will explore and revise fundamental concepts of vectors, as well as how vector algebra can be applied to three dimensional geometry. Students will also be introduced to direction, electronic direction cosines, and direction ratios when a line is joined to two points. Additionally, this chapter will cover the equations of line and plane in space under various conditions, the angles between lines, planes, and planes, the shortest distances between skew lines, and the distance from a point to a plane.

11.2 Direction Cosines and Direction Ratios of a line

1. Direction Cosines of a line passing through two Points

In chapter 11 of class 12 mathematics, students will be taught and observed the direction cosine and direction ratio of a line using an example as a starting point. Subsequently, they will be taught how the line does not traverse the origin of the space, and how to find the direction cosine by drawing a line through the origin of the line and parallel to it. You'll also look at the direction cosine of a line that runs between two points.

11.3 Equation of a line in Space

1. Equation of a line through a given point and Parallel to “a” given vector “b”

In the class 12 solutions for 3D geometry, you'll look at how a line can be uniquely defined if it goes through a certain point and has a direction, or if it goes through two different points. You'll also look at how the lines in space can be represented using vector equations and cartesian integrals. In this section, further, you will be introduced to the equation between a line passing through a point and running parallel to a vector.

11.4 Angle between two lines - In this section, an example will be provided prior to the introduction of the concept of Angle between Two lines in Three-dimensional Geometry.

11.5 Shortest Distance between two lines

1. Distance between two Skew lines

2. Distance between Parallel lines

Here, you'll learn how if two lines are parallel to each other, the shortest distance is the perpendicular distance, which is the distance from one point on one line to the other. You'll also learn how and why this is the case. Basically, it means that if you join one point in one line to another point on the same line, you get the shortest distance. In addition, you will learn how the shortest distance for skew lines is perpendicular to both lines. Further, you will learn about the distance from two skew lines, and the distance from parallel lines.

## NCERT Solutions for Class 12 Maths Chapter 11 Exercises

NCERT solutions for chapter 11 maths class 12 exercises provide a comprehensive and organized approach to solving numerical problems. These solutions are carefully designed by subject matter specialists, taking into account the learning needs and abilities of students.

NCERT solutions for chapter 11 of maths for class 12 include:

 Exercise 11.1 5 Questions & Solutions Exercise 11.2 15 Questions & Solutions Miscellaneous Exercise 5 Questions & Solutions

Chapter 11 is concerned with the exploration of three-dimensional points, lines and planes, This enables a better understanding of spatial concepts. To gain a better understanding of this difficult topic, students can benefit greatly from downloading the NCERT solutions for class 12 chapter 11.

Here, we'll look at some of the main benefits of using these solutions.

1. These solutions are accompanied by pertinent clarifications and illustrations.

2. These solutions have been created by subject experts, and it's been thoroughly researched and organized to provide comprehensive solutions to every question in NCERT chapter 11.

3. With the NCERT solutions, you can do lots of exercises and examples to help you stay on top of your learning and make sure you're up to date with the concepts.

4. With organized solutions provided by eSaral, students can solve questions quickly and use their time wisely on different parts of their question paper.

5. Through consistent practice and a clear comprehension of the concepts through these solutions, students gain a sense of self-confidence.

Question 1. What is Three Dimensional Geometry?

Answer 1. The field of three-dimensional geometry is a branch of mathematics concerned with the investigation of points, rows, or solid forms in a system of three-dimensional coordinates. This area of mathematics will introduce the notion of Z-coordinate, as well as the x- and y-coordinates for determining the exact position of a point within the 3-dimensional coordinate plane.

Question 2.  How many exercises are there in chapter 11?

Answer 2. There are 3 exercises in class 12 maths in chapter 11.

1. Exercise 11.1

2. Exercise 11.2

3. Miscellaneous Exercise