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( x₁ , y ₁ , z₁ ) and B( x₂ , y₂ , z₂ ): √[(x₂– x₁ )² + (y₂ – y₁)² + (z₂ – z₁)² ]
• The coordinates of the mid-point of the line segment which joins two points A( x₁ , y ₁ , z₁ ) and B( x₂ , y₂ , z₂ ) are [ (x₁ + x₂ )/2, (y₁ + y₂ /2) , (z₁ + z₂ /2) ]
• The coordinates of the centroid of the triangle, whose vertices are (x₁ , y ₁ , z₁ ) , (x₂ , y₂ , z₂ ) and (x₃ , y₃ , z ₃ ) are [ (x₁ + x₂ + x₃ )/3, (y₁ + y₂+ y₃ /3) , (z₁ + z₂ + z₃ /3) ]