Three Dimensional Geometry NCERT Solutions Class 12 Maths Chapter 11 Exercise 11.1 11.2 11.3 Miscellaneous Free pdf Notes Study Material download-Anand Classes
NCERT Solutions for Class 12 Maths – Chapter 11 – Three Dimensional Geometry– is designed and prepared by the best teachers of ANAND CLASSES. All the important topics are covered in the exercises and each answer comes with a detailed explanation to help students understand concepts better. These NCERT solutions play a crucial role in your preparation for all exams conducted by the CBSE, including the JEE.
Chapter 11 – Three Dimensional Geometry covers multiple exercises. The answer to each question in every exercise is provided along with complete, step-wise solutions for your better understanding. This will prove to be most helpful to you in your home assignments as well as practice sessions.
Download the pdf of NCERT Solutions for Class 12 Maths Chapter 11 – Three Dimensional Geometry
The major concepts of Maths covered in Chapter 11- Three Dimensional Geometry of NCERT Solutions for Class 12 are:
- 11.1 Introduction
- 11.2 Direction Cosines and Direction Ratios of a Line
- 11.2.1 Relation between the direction cosines of a line
- 11.2.2 Direction cosines of a line passing through two points
- 11.3 Equation of a Line in Space
- 11.3.1 Equation of a line through a given point and parallel to a given vector b.
- 11.3.2 Equation of a line passing through two given points
- 11.4 Angle between Two Lines
- 11.5 Shortest Distance between Two Lines
- 11.5.1 Distance between two skew lines
- 11.5.2 Distance between parallel lines
- 11.6 Plane
- 11.6.1 Equation of a plane in normal form
- 11.6.2 Equation of a plane perpendicular to a given vector, passing through a given point
- 11.6.3 Equation of a plane passing through three non-collinear points
- 11.6.4 Intercept form of the equation of a plane
- 11.6.5 Plane passing through the intersection of two given planes
- 11.7 Coplanarity of Two Lines
- 11.8 Angle between Two Planes
- 11.9 Distance of a Point from a Plane
- 11.10 Angle between a Line and a Plane
NCERT Solutions for Class 12 Maths Chapter 11 – Three Dimensional Geometry
Three Dimensional Geometry chapter belongs to the Unit Vectors and its solutions are prepared as per latest CBSE Syllabus. It adds up to 14 marks of the total marks. There are 3 exercises along with a miscellaneous exercise in this chapter to help students understand the concepts related to Three Dimensional Geometry clearly. Some of the topics discussed in Chapter 11 of NCERT Solutions for Class 12 Maths are as follows:
- Direction cosines of a line are the cosines of the angles made by the line with the positive directions of the coordinate axes.
- If l, m, n are the direction cosines of a line, then l2 + m2 + n2 = 1
- Direction ratios of a line are the numbers which are proportional to the direction cosines of a line.
- Skew lines are lines in space which are neither parallel nor intersecting. They lie in different planes.
- Angle between skew lines is the angle between two intersecting lines drawn from any point (preferably through the origin) parallel to each of the skew lines.
- If l1 , m1, n1 and l2, m2, n2 are the direction cosines of two lines; and θ is the acute angle between the two lines; then cosθ = |l1l2 + m1m2 + n1n2|
These are a few topics that are discussed in chapter three dimensional geometry. Students can utilise the NCERT Solutions for Class 12 Maths Chapter 11 for any quick reference to comprehend complex topics. Also, to know more about the chapter, refer to the NCERT Textbook and NCERT Solutions of Class 12 Maths.
Features of NCERT Solutions for Class 12 Maths Chapter 11 – Three Dimensional Geometry
Studying the Three Dimensional Geometry of Class 12 using the NCERT Solutions enables the students to understand the following:
- Direction cosines and direction ratios of a line joining two points.
- Cartesian equation and vector equation of a line, coplanar and skew lines, the shortest distance between two lines.
- Cartesian and vector equation of a plane.
- The angle between (i) two lines, (ii) two planes, (iii) a line and a plane.
- Distance of a point from a plane.
Frequently Asked Questions (FAQs) on NCERT Solutions for Class 12 Maths Chapter 11
What is three dimensional geometry in NCERT Solutions for Class 12 Maths Chapter 11?
Three dimensional geometry is a branch of Mathematics that includes the study of lines, points, and solid shapes in three dimensional coordinate systems. It will introduce students to the concept of z-coordinate along with x, y and z coordinates to determine the exact location of a point in the three dimensional coordinate planes.
What are the important topics covered in Chapter 11 of NCERT Solutions for Class 12 Maths?
11.2 Direction Cosines and Direction Ratios of a Line
11.3 Equation of a Line in Space
11.4 Angle between Two Lines
11.5 Shortest Distance between Two Lines
11.7 Coplanarity of Two Lines
11.8 Angle between Two Planes
11.9 Distance of a Point from a Plane
11.10 Angle between a Line and a Plane
Is NCERT Solutions for Class 12 Maths Chapter 11 the best study material for the students during revision?
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