CBSE Class 11 Mathematics Chapter 12: Introduction to Three Dimensional Geometry NCERT Solutions

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NCERT Solutions PDF Class 11 PDF

This chapter introduces the fundamental concepts of three-dimensional geometry for Class 11 CBSE students. It explains the coordinate system in 3D space, including the three mutually perpendicular axes (x, y, and z) and the coordinate planes (XY, YZ, and XZ planes). The solutions detail how these planes divide space into eight octants and how to determine the octant a point lies in based on the signs of its coordinates. It also covers the characteristics of points lying on axes or coordinate planes. These NCERT Solutions provide clear, step-by-step explanations to help students grasp the basics of 3D geometry, essential for understanding more advanced topics in mathematics and physics. They are ideal for exam preparation, offering a solid foundation for visualizing and working with points and planes in three dimensions.

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Quick info

BoardCBSE
ClassClass 11
SubjectMathematics
Session2026
LanguageEnglish
TypeNCERT Solutions
ChapterChapter 12: Introduction to Three Dimensional Geometry

Chapter summary

Chapter 12, Introduction to Three Dimensional Geometry, lays the groundwork for 3D coordinate systems. This NCERT Solutions set focuses on understanding the relationship between points and their coordinates in 3D space. It covers the definition and properties of the x, y, and z axes, the coordinate planes, and the eight octants. The solutions explain how to identify the octant of a given point and the form of coordinates for points lying on axes or planes.

Learning outcomes

  • Understand the concept of three-dimensional coordinate system.
  • Identify the coordinate planes and axes in 3D space.
  • Determine the octant in which a point lies based on its coordinates.
  • Describe the coordinates of points lying on the axes or coordinate planes.
  • Visualize points and planes in three-dimensional space.

Topics covered

Paper topics

  • Three-dimensional coordinate system
  • Axes (x-axis, y-axis, z-axis)
  • Coordinate planes (XY-plane, YZ-plane, XZ-plane)
  • Octants
  • Coordinates of a point in space
  • Points on axes
  • Points on coordinate planes
  • Signs of coordinates in different octants

Important topics

  • Understanding the octants and their sign conventions
  • Determining the octant of a given point
  • Properties of points lying on axes
  • Properties of points lying on coordinate planes

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Questions and Solutions

Question 1

A point is on the x-axis. What are its y-coordinates and z-coordinates?
Solution:

In a three-dimensional coordinate system, the x-axis is defined as the line where both the y-coordinate and the z-coordinate are zero. Therefore, any point that lies on the x-axis will have its y-coordinate and z-coordinate equal to 0. The coordinates of such a point can be represented as (x, 0, 0), where 'x' can be any real number.

Question 2

A point is in the XZ-plane. What can you say about its y-coordinate?
Solution:

The three coordinate planes are the XY-plane, the YZ-plane, and the XZ-plane. These planes are formed by taking pairs of axes. The XZ-plane is specifically defined as the set of all points where the y-coordinate is equal to zero. Thus, if a point lies in the XZ-plane, its y-coordinate must be 0.

Question 3

Name the octants in which the following points lie: (1, 2, 3), (4, -2, 3), (4, -2, -5), (4, 2, -5), (-4, 2, -5), (-4, 2, 5), (-3, -1, 6), (2, -4, -7)
Solution:

The octants in a 3D coordinate system are determined by the signs of the x, y, and z coordinates. Here's how we determine the octant for each point:

  • Point (1, 2, 3): All coordinates (x, y, z) are positive. This point lies in Octant I.
  • Point (4, -2, 3): The signs are (+, -, +). This combination corresponds to Octant IV.
  • Point (4, -2, -5): The signs are (+, -, -). This combination corresponds to Octant VIII.
  • Point (4, 2, -5): The signs are (+, +, -). This combination corresponds to Octant V.
  • Point (-4, 2, -5): The signs are (-, +, -). This combination corresponds to Octant VI.
  • Point (-4, 2, 5): The signs are (-, +, +). This combination corresponds to Octant II.
  • Point (-3, -1, 6): The signs are (-, -, +). This combination corresponds to Octant III.
  • Point (2, -4, -7): The signs are (+, -, -). This combination corresponds to Octant VIII.

Question 4

Fill in the blanks: (i) The x-axis and y-axis taken together determine a plane known as_____. (ii) The coordinates of points in the XY-plane are of the form _____. (iii) Coordinate planes divide the space into _____ octants.
Solution:

(i) The x-axis and y-axis, being perpendicular to each other, define a plane. This plane is known as the xy-plane (or XY-plane).

(ii) The XY-plane is the set of all points where the z-coordinate is zero. Therefore, the coordinates of any point lying in the XY-plane are of the form (x, y, 0).

(iii) The three coordinate planes (XY-plane, YZ-plane, and XZ-plane) are mutually perpendicular and intersect at the origin. They divide the entire three-dimensional space into eight distinct regions, which are called octants.

Common mistakes

  • Confusing the order of coordinates (x, y, z).
  • Incorrectly determining the octant based on the signs of coordinates.
  • Misunderstanding the properties of points lying on axes or planes.

Revision tips

  • Draw a 3D coordinate system and plot sample points to visualize octants.
  • Memorize the sign conventions for each octant.
  • Practice identifying the coordinates of points on axes and planes.
  • Review the definitions of coordinate planes and axes thoroughly.

Practice MCQs

Q1. What are the y and z coordinates of a point lying on the x-axis?

Q2. If a point lies in the XZ-plane, what is its y-coordinate?

Q3. In which octant does the point (4, -2, 3) lie?

Q4. A point has coordinates of the form (x, y, 0). Where does this point lie?

Q5. How many octants divide the 3D space by the coordinate planes?

Frequently asked questions

What is the main concept of Chapter 12: Introduction to Three Dimensional Geometry?

This chapter introduces the basic framework of a three-dimensional coordinate system, including the x, y, and z axes, the coordinate planes, and how these divide space into eight octants. It focuses on locating points in 3D space.

How do I determine which octant a point lies in?

You determine the octant by looking at the signs of the x, y, and z coordinates. Each of the eight octants is defined by a unique combination of positive and negative signs for the coordinates.

What are the coordinates of a point on the x-axis?

A point on the x-axis has its y-coordinate and z-coordinate equal to zero. Its coordinates are of the form (x, 0, 0).

What is the significance of the XY-plane?

The XY-plane is one of the three coordinate planes. Any point lying on the XY-plane has a z-coordinate of zero, and its coordinates are of the form (x, y, 0).

How can these NCERT Solutions help with exam preparation?

These solutions provide clear, step-by-step explanations for each problem, reinforcing the understanding of 3D geometry concepts. Practicing these problems helps in visualizing points in space and correctly identifying their locations, which is crucial for exams.

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