CBSE Class 11 Mathematics Chapter 12: Introduction to Three Dimensional Geometry NCERT Solutions
This chapter introduces the fundamental concepts of three-dimensional (3D) geometry. Students will learn about the coordinate system in 3D space, including the x-axis, y-axis, and z-axis, and how they intersect at the origin. The solutions explain the division of space into eight octants based on the signs of the coordinates. Key topics covered include identifying the octant a point lies in, understanding points on coordinate axes and planes, and filling in blanks related to these concepts. These NCERT Solutions provide clear, step-by-step explanations to help students grasp the basics of 3D geometry, which is crucial for further studies in mathematics and physics. They are designed to aid in exam preparation by reinforcing understanding of coordinate systems and spatial visualization.
Quick info
| Board | CBSE |
|---|---|
| Class | Class 11 |
| Subject | Mathematics |
| Session | 2026 |
| Language | English |
| Type | NCERT Solutions |
| Chapter | Chapter 12 |
Chapter summary
Chapter 12, Introduction to Three Dimensional Geometry, lays the groundwork for understanding points and their locations in 3D space. This NCERT Solutions set focuses on Exercise 12.1, clarifying concepts like coordinate axes, coordinate planes (XY, YZ, XZ planes), and the eight octants. It provides detailed answers for identifying octants based on coordinate signs and understanding the form of coordinates on axes and planes.
Learning outcomes
- Understand the concept of a point in three-dimensional space.
- Identify the coordinate planes (XY, YZ, XZ) and coordinate axes (x, y, z).
- Determine the octant in which a given point lies based on the signs of its coordinates.
- Describe the coordinates of a point lying on a coordinate axis or a coordinate plane.
- Recall the total number of octants formed by the coordinate planes.
Topics covered
Paper topics
- Three-dimensional coordinate system
- Coordinate axes (x-axis, y-axis, z-axis)
- Coordinate planes (XY-plane, YZ-plane, XZ-plane)
- Origin
- Octants
- Coordinates of a point
- Points on axes
- Points on coordinate planes
- Signs of coordinates in octants
- Introduction to 3D Geometry
Important topics
- Understanding and naming the eight octants
- Determining the octant of a point based on coordinate signs
- Coordinates of points on axes and planes
- The role of coordinate planes in dividing space
PDF preview
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Questions and Solutions
Question 1
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 lying on the x-axis will have its y-coordinate and z-coordinate equal to 0. The coordinates of such a point are of the form .
Question 2
The XZ-plane is one of the three coordinate planes that divide the 3D space. This plane is formed by the x-axis and the z-axis. By definition, all points lying in the XZ-plane have a y-coordinate equal to zero. This is because the XZ-plane represents the set of all points with no displacement along the y-axis. Thus, if a point is in the XZ-plane, its y-coordinate must be 0.
Question 3
The three coordinate planes (XY, YZ, XZ) divide the 3D space into eight regions called octants. The octant in which a point lies is determined by the signs of its x, y, and z coordinates. Here is the breakdown 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 x-coordinate is positive, the y-coordinate is negative, and the z-coordinate is positive. This combination corresponds to Octant IV.
- Point (4, -2, -5): The x-coordinate is positive, the y-coordinate is negative, and the z-coordinate is negative. This point lies in Octant VIII.
- Point (4, 2, -5): The x-coordinate is positive, the y-coordinate is positive, and the z-coordinate is negative. This point lies in Octant V.
- Point (-4, 2, -5): The x-coordinate is negative, the y-coordinate is positive, and the z-coordinate is negative. This point lies in Octant VI.
- Point (-4, 2, 5): The x-coordinate is negative, the y-coordinate is positive, and the z-coordinate is positive. This point lies in Octant II.
- Point (-3, -1, 6): The x-coordinate is negative, the y-coordinate is negative, and the z-coordinate is positive. This point lies in Octant III.
- Point (2, -4, -7): The x-coordinate is positive, the y-coordinate is negative, and the z-coordinate is negative. This point lies in Octant VIII.
Question 4
(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.
(i) The x-axis and y-axis are two of the three coordinate axes. When taken together, they define the XY-plane, which is one of the three coordinate planes.
(ii) The XY-plane is the plane containing the x-axis and the y-axis. Any point lying on this plane has a z-coordinate of 0. Therefore, the coordinates of points in the XY-plane are of the form .
(iii) The three mutually perpendicular coordinate planes (XY-plane, YZ-plane, and XZ-plane) divide the entire three-dimensional space into exactly eight regions, which are known as octants.
Common mistakes
- Confusing the order of coordinates (x, y, z) when determining the octant.
- Incorrectly identifying the signs of coordinates in different octants.
- Mistaking points on axes for points on planes, or vice versa.
- Forgetting that the coordinate planes divide space into exactly eight octants.
Revision tips
- Visualize the 3D coordinate system: Imagine the x, y, and z axes forming a corner like in a room.
- Memorize the sign conventions for each octant. A table can be very helpful.
- Practice plotting points mentally or on paper to reinforce octant identification.
- Review the definitions of coordinate axes and coordinate planes thoroughly.
Practice MCQs
Q1. What are the y and z coordinates of a point lying on the x-axis?
Explanation: A point on the x-axis has no displacement along the y or z directions, hence its y and z coordinates are both zero.
Q2. A point lies in the XZ-plane. What is its y-coordinate?
Explanation: The XZ-plane is defined by all points where the y-coordinate is zero. This plane separates points with positive y-coordinates from those with negative y-coordinates.
Q3. In which octant does the point (4, -2, 3) lie?
Explanation: The point (4, -2, 3) has a positive x-coordinate, a negative y-coordinate, and a positive z-coordinate. This combination corresponds to Octant IV.
Q4. The coordinates of a point in the XY-plane are of the form:
Explanation: The XY-plane is the plane formed by the x-axis and y-axis. Any point in this plane has a z-coordinate of zero.
Q5. How many octants are created by the three coordinate planes?
Explanation: The three mutually perpendicular coordinate planes (XY, YZ, XZ) divide the entire three-dimensional space into exactly eight regions, known as octants.
Frequently asked questions
What is the main concept covered in Chapter 12, Exercise 12.1 of CBSE Class 11 Maths?
Chapter 12, Exercise 12.1, introduces the basic concepts of three-dimensional geometry, focusing on the coordinate system, coordinate axes, coordinate planes, and the division of space into eight octants.
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 octant is defined by a unique combination of positive and negative signs for these coordinates.
What are the coordinates of a point on the x-axis?
A point on the x-axis has its y and z coordinates equal to zero. Its coordinates are of the form (x, 0, 0).
What is the significance of the XZ-plane?
The XZ-plane is one of the three coordinate planes. Any point lying on the XZ-plane has a y-coordinate of zero, meaning it has no displacement along the y-axis.
How many octants are there in a 3D coordinate system?
The three mutually perpendicular coordinate planes divide the 3D space into exactly eight regions, called octants.
How can these NCERT solutions help with exam preparation?
These solutions provide clear, step-by-step explanations for each question in Exercise 12.1, helping students understand the fundamental concepts of 3D geometry and reinforcing their problem-solving skills for exams.
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