## Introduction To Three Dimensional Geometry |

# Three Dimensional Geometry NCERT Solutions Class 11 Maths Chapter 12 Exercise 12.1 12.2 12.3 Miscellaneous Free pdf Notes Study Material download-Anand Classes

NCERT Solutions for Class 11 Maths Chapter 12 Introduction to Three Dimensional Geometry goes through the fundamentals of geometry in three dimensions. Students must be familiar by now with the basics of coordinate geometry, such as in order to locate any point in space, we need two intersecting lines, also called coordinate axes, and the information about the two numbers known as the coordinates of the point with respect to the axes. But, what if we want to find the position of an object in real life, then three-dimensional geometry comes into play and we need a third reference. This chapter will introduce geometry in three dimensions where we will require a third factor, i.e., the height of the object from the ground. Thus, these NCERT solutions Class 11 maths Chapter 12 explain the coordinate axes and the coordinate planes in three-dimensional space.

# Download the pdf of NCERT Solutions for Class 11 Maths Chapter 12 – Introduction to Three Dimensional Geometry

# Download the NCERT Book for Class 11 Maths Chapter 12 – Introduction to Three Dimensional Geometry

# Number of questions in each exercise of NCERT Solutions for Class 11 Maths Chapter 12 – Introduction to Three Dimensional Geometry

With relevant practical examples, the students will be able to grasp geometrical forms and patterns by combining the power of algebra with coordinate geometry. They will get to learn the practical utility of the three coordinates, and how this can be used to find out measures like point distance and angle between lines, which are the foundation for understanding shapes like triangles and parallelograms. The Class 11 maths NCERT solutions Chapter 12 Introduction To Three Dimensional Geometry pdf files are given below for reference and also download the exercise-wise solutions provided in the links below.

- NCERT Solutions Class 11 Maths Chapter 12 Ex 12.1
- NCERT Solutions Class 11 Maths Chapter 12 Ex 12.2
- NCERT Solutions Class 11 Maths Chapter 12 Ex 12.3
- NCERT Solutions Class 11 Maths Chapter 12 Miscellaneous Exercise

The subject of geometry cannot be ignored because it’s all around us. One of the most renowned applications of three-dimensional geometry is in the Space-Time Continuum of Einstein’s Theory of Relativity. In order to solve any problem involving a specific shape, it is critical to understand how to measure its dimensions or recognize it.

The branch of mathematics that deals with spatial interactions is known as geometry. It also assists us in finding distances and heights in our everyday lives. From evaluating the capacity of a box to determining the size of a room, the use of three-dimensional geometry is inevitable. As a result, it is critical for the students to understand the concepts of three-dimensional space. An exercise-wise detailed analysis of NCERT Solutions Class 11 Maths Chapter 12 Introduction To Three Dimensional Geometry is given below :

- Class 11 Maths Chapter 12 Ex 12.1 – 4 Questions
- Class 11 Maths Chapter 12 Ex 12.2 – 5 Questions
- Class 11 Maths Chapter 12 Ex 12.3 – 5 Questions
- Class 11 Maths Chapter 12 Miscellaneous Exercise

# Important Topics and Subtopics of NCERT Solutions for Class 11 Maths Chapter 12 – Introduction to Three Dimensional Geometry

**Topics Covered:** The Class 11 maths NCERT solutions Chapter 12 covers the topics like how coordinates of a point in space are identified, how the distance between two points is calculated, the section formula, and how to prove the collinearity of points using the section formula.

**Total Questions:** The Class 11 Maths Chapter 12 Introduction To Three Dimensional Geometry Chapter 12 consists of 14 questions of which 9 are easy, 3 are moderately simple and 3 are long answer-type questions.

Given below are the topics of Class 11 Maths Chapter 12, Introduction to Three Dimensional Geometry, which is categorised under the CBSE Class 11 Maths Syllabus.

**12.1 Introduction**

This section introduces the concept of coordinate axes, coordinate planes in real life, coordinates of the point concerning the three coordinate planes, and the basics of geometry in three-dimensional space.

**12.2 Coordinate Axes and Coordinate Planes in Three Dimensional Space**

This section defines the rectangular coordinate system, the naming of a coordinate plane and different notations in coordinate planes.

Rahul was riding his bicycle back home from a basketball game at a nearby stadium when he hit the divider to avoid a dog that had run onto the road. Unfortunately, his bicycle was stuck to the divider. When he couldn’t remove the bicycle by himself, he decided to take the help of his friend who stayed nearby. Rahul later looked at a topographical map and identified his friend’s house. He travelled 200 metres south and 550 metres west from where he left his bicycle. The map showed that he had also walked uphill from an altitude of 600 metres to an altitude of 650 metres above sea level. If we treat the location of Rahul’s bicycle as the origin of the coordinates, what is the position vector of the Kapur farm?

**12.3 Coordinates of a Point in Space**

This section explains the coordinate system in space, coordinates [x, y and z] with a few examples.

If a student is planning to place different pieces of furniture in a drawing room, a two-dimensional grid representing the room can be drawn, and an appropriate unit of measurement should be used. Let the horizontal distance be *x*, the vertical distance to x be *y*, and the origin is the starting point. If the width of the room is 10 meters, any point in the room can be defined as (x,y,z).

**12.4 Distance between Two Points**

This section covers the distance between three points in a three dimensional coordinate system using the distance formula along with a few solved problems.

If an object P is placed in the coordinate plane, the distance between the point P from the three axes [x, y and z] can be calculated using the distance formula.

**12.5 Section Formula**

This section talks about the section formula for a three dimensional geometry and its different cases. A few examples are solved for better understanding.

- Exercise 12.1 Solutions: 4 Questions
- Exercise 12.2 Solutions: 5 Questions
- Exercise 12.3 Solutions: 5 Questions
- Miscellaneous Exercise on Chapter 12 Solutions: 6 Questions

## Important Points on Chapter 12 Introduction to Three Dimensional Geometry

- In three dimensional geometry, a cartesian coordinate system consists of three mutually perpendicular lines, namely x, y and z-axes. They are measured in the same unit of length.
- The three planes, XY-plane, YZ-plane and ZX-plane, are determined by the pair of axes called the axes of the coordinate planes.
- The three coordinate planes divide the space into eight parts known as octants.
- The coordinates of a point P (x, y, z) in three dimensional geometry are written in the form of an ordered triplet. Here, x, y and z are the distances from the YZ, ZX and XY-planes.
- (i) Any point on the x-axis is represented as (x, 0, 0), (ii) Any point on the y-axis is represented as (0, y, 0), and (iii) Any point on the z-axis is represented as (0, 0, z).
- The coordinates of point R divide the line segment joining two points P (x1,y1,z1) and Q (x2,y2,z2) internally and externally in the ratio m:n.

# List of Formulas in NCERT Solutions Class 11 Maths Chapter 12 – Introduction to Three Dimensional Geometry

NCERT solutions Class 11 maths Chapter 12 explains the three-dimensional space in the cartesian coordinate system using the x, y, and z axes. The position of any point in space can be determined using these coordinates. It also presents us with a lot more information when combined with other factors by using certain formulas. This chapter requires the extensive use of formulas. In the absence of these formulas determining, say, the centroid of a triangle, would take days rather than mere minutes. Thus, they form an important component of the NCERT solutions for Class 11 Maths Chapter 12. A few of them are given below:

**Distance between any two points A( x**_{1 }, y_{1 }, z_{1}) and B( x_{2 }, y_{2 }, z_{2}):**√[****(x**_{2}– x_{1 })^{2}+ (y_{2}– y_{1})^{2}+ (z_{2}– z_{1})^{2}]**The coordinates of the mid-point of the line segment which joins two points A( x**_{1 }, y_{1 }, z_{1}) and B( x_{2 }, y_{2 }, z_{2}) are [ (x_{1}+ x_{2})/2, (y_{1}+ y_{2}/2) , (z_{1}+ z_{2}/2) ]**The coordinates of the centroid of the triangle, whose vertices are (x**_{1 }, y_{1 }, z_{1}) , (x_{2 }, y_{2 }, z_{2}) and (x_{3}, y_{3}, z_{ 3}) are [ (x_{1}+ x_{2}+ x_{3})/3, (y_{1}+ y_{2}+ y_{3}/3) , (z_{1}+ z_{2}+ z_{3}/3) ]

# Frequently Asked Questions (FAQs) on NCERT Solutions Class 11 Maths Chapter 12 – Introduction to Three Dimensional Geometry

**Q1**

### Why are NCERT Solutions Class 11 Maths Chapter 12 Important?

The examples and exercises presented in the NCERT Solutions Class 11 Maths Chapter 12 are based on real-life activities; they teach practical information in addition to the necessary theoretical knowledge. The principles covered in NCERT Solutions are useful for strengthening exam study skills. As a result, these solutions are a valuable resource for learning about the various dimensions and characteristics of three-dimensional geometry.

**Q2**

### Do I Need to Practice all Questions Provided in NCERT Solutions Class 11 Maths Introduction To Three Dimensional Geometry?

NCERT’s team ensures that the content gathered in NCERT Solutions Class 11 Maths Introduction To Three Dimensional Geometry is easy to understand and covers all of the major concepts. All of the exercises, examples, and exercise problems have been carefully crafted to assist students in getting a quick and effective understanding of the various topics. As a result, students would benefit from making use of this fantastic resource and by practicing all of the questions in order to assimilate the fundamental principles of three-dimensional space.

**Q3**

### Why Should I Practice NCERT Solutions Class 11 Maths Introduction To Three Dimensional Geometry Chapter 12?

The NCERT solutions are prepared by eminent scholars after a lot of research, and also their intention is to present the matter in an easily understandable language. The CBSE board too highly recommends these solutions for study; hence the probability of the questions landing up in the board examinations is high. Also, the students can learn a strategic approach towards problem-solving by referring to the NCERT Solutions Class 11 Maths Introduction To Three Dimensional Geometry Chapter 12.

**Q4**

### What are the Important Topics Covered in Class 11 Maths NCERT Solutions Chapter 12?

The important topics covered in the NCERT Solutions for Class 11 math Chapter 12 includes a description of the rectangular Cartesian coordinate system, how to identify the coordinates of a location in space, and how to compute the distance between two points. What is the section formula, and how may it be used to verify point collinearity, the formula for finding the coordinates of the mid-point of the line segment, and to find the coordinates of the centroid of the triangle, are also explained in detail.

**Q5**

### How Many Questions are there in NCERT Solutions Class 11 Maths Chapter 12 Introduction To Three Dimensional Geometry?

The NCERT Solutions Class 11 Maths Chapter 12 Introduction To Three Dimensional Geometry Chapter 12 has 14 questions, of which 9 are easy, 3 are moderately easy, and 3 are long answer type questions. There is also a miscellaneous exercise at the end of the chapter whose problems combine all of the concepts learned in the chapter. It has 5 questions, which will be advantageous for the students to learn higher-level math.

**Q6**

### How CBSE Students can utilize NCERT Solutions Class 11 Maths Chapter 12 effectively?

In the NCERT solutions, there are several activities and examples that students should practice because this will help them build the necessary logic to understand the chapter well. After that, they must solve all of the exercise problems. Each activity, example, and problem question will assist students in exploring various aspects of three-dimensional geometry, allowing them to use the NCERT Solutions Class 11 Maths Chapter 12 effectively.

**Q7**

### Discuss the topics covered in Chapter 12 of NCERT Solutions for Class 11 Maths.

1. Introduction

2. Coordinate Axes and Coordinate Planes in Three Dimensional Space

3. Coordinates of a Point in Space

4. Distance between Two Points

5. Section Formula

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