# NCERT Solutions Class 12 Maths Chapter 11 Three Dimensional Geometry

NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry explains the vector algebra of three-dimensional geometry. This chapter introduces the concept of z-coordinate along with x and y coordinates for students to understand and determine the location of points in a three-dimensional plane. The focus is on studying the direction cosines and direction ratios of a line joining two points. It also explains the equations of lines and planes in space under different conditions, the angle between two lines, two planes, a line, and a plane, shortest distance between two skew lines, and distance of a point from a plane. Three Dimensional Geometry is an essential part of math studies thus, attaining an in-depth understanding of this topic is highly crucial. NCERT Solutions Class 12 Maths Chapter 11 helps students to acquire a fundamental understanding of Three Dimensional Geometry.

Some of the important concepts explained in this chapter are based on the fundamentals of the cartesian and vector equation of a line in three-dimensional geometry. With the regular practice of Class 12 Maths NCERT Solutions Chapter 11, students will quickly gain a deep understanding of this topic to perform best in exams. To study the core basics of three-dimensional geometry, download the exercises provided in the links below.

- NCERT Solutions Class 12 Maths Chapter 11 Ex 11.1
- NCERT Solutions Class 12 Maths Chapter 11 Ex 11.2
- NCERT Solutions Class 12 Maths Chapter 11 Ex 11.3
- NCERT Solutions Class 12 Maths Chapter 11 Miscellaneous Exercise

## NCERT Solutions for Class 12 Maths Chapter 11 PDF

Three-dimensional geometry is the branch of mathematics that involves studying the properties of lines, points, and various solid shapes in three-dimensional coordinate systems. NCERT Solutions for Class 12 Maths Chapter 11 are well-crafted guides formulated to promote the understanding of these topics and how to apply them to varying levels of questions. To practice with these solutions, click on the links of the pdf files given below.

**ā** Download Class 12 Maths NCERT Solutions Chapter 11 Three Dimensional Geometry

**NCERT Class 12 Maths Chapter 12 **

### NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry

NCERT solutions Class 12 Maths Chapter 11 Three Dimensional Geometry are highly competent resources devised in an efficient manner to impart precise geometry skills in students. These solutions incorporate all topics and concepts related to using vector algebra in three-dimensional geometry. These reliable learning resources are efficient to strengthen the core basics. The vernacular language used in these solutions makes it quite simple to understand all topics clearly. To practice the exercise-wise NCERT Solutions Class 12 Maths, try the links given below.

- Class 12 Maths Chapter 11 Ex 11.1 - 5 Questions
- Class 12 Maths Chapter 11 Ex 11.2 - 17 Questions
- Class 12 Maths Chapter 11 Ex 11.3 - 14 Questions
- Class 12 Maths Chapter 11 Miscellaneous Ex - 23 Questions

ā Download CBSE Class 12 Maths Chapter 11 NCERT Book

**Topics Covered:** The important topics covered in NCERT Solutions Class 12 Maths Chapter 11 are related to the direction cosines and direction ratios of a line, equation of a line in space, the angle between two lines, the shortest distance between two lines, plane, coplanarity of two lines, the angle between two planes, the distance of a point from a plane, the angle between a line and a plane.

**Total Questions:** Class 12 Maths Chapter 11 Three Dimensional Geometry has 36 questions in 3 exercises and 23 more provided in a miscellaneous exercise. Out of these 59 questions, there are long answers, short answers and MCQs provided in each exercise to cover all important topics from an exam point of view.

## List of Formulas in NCERT Solutions Class 12 Maths Chapter 11

Three Dimensional Geometry is an important chapter in class 12th maths that carries a significant weightage of around 14 marks. The simple way to score well on this topic is to prepare all core concepts and formulas properly. NCERT Solutions Class 12 Maths Chapter 11 elaborates each concept, formula, term, and definition with the help of interactive illustrations for students to form a firm base. Some of the important terms and definitions explained in these NCERT Solutions for Class 12 Maths Chapter 11 are given below.

**Direction cosines of a line:**These 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 l
^{2 }+ m^{2}+ n^{2}= 1.

**Direction ratios of a line:**Any three numbers proportional to the direction cosines of a line are called the direction ratios of the line.

**Skew lines:**Lines in space that are neither parallel nor intersecting are called skew lines. These lines lie in different planes.

**Angle between skew lines:**Angle between two skew lines is the angle between two intersecting lines drawn from any point parallel to each of the skew lines.

- Shortest distance between two skew lines is the line segment perpendicular to both the lines.

- If l
_{1}, m_{1}, n_{1}and l_{2}, m_{2}, n_{2}are the direction cosines of two lines; and θ is the acute angle between the two lines; then cosθ = |l_{1}l_{2}+ m_{1}m_{2}+ n_{1}n_{2}|

**Cartesian equation of a plane:**The Cartesian form of the equation of the plane is lx + my + nz = d.

**Angle between two planes:**The angle between two planes is defined as the angle between their normals.

## FAQs on NCERT Solutions Class 12 Maths Chapter 11

### What is the Importance of NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry?

NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry are efficiently designed by maths experts to impart a deep knowledge of all basics. Strategically preparing all the important topics with the help of these resources will enable students to attain a thorough conceptual understanding to perform well in exams. The well-crafted format of these solutions is excellent to encourage math practice which in turn helps in enhancing the confidence of students.

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

Chapter 11 Three Dimensional Geometry of class 12 Maths briefly introduces all the important concepts of three-dimensional geometry. The important topics covered in the NCERT Solutions Class 12 Maths Chapter 11 are direction cosines, direction ratios of a line joining two points, cartesian equation of a line, vector equation of a line, coplanar lines, skew lines, the shortest distance between two lines, cartesian and vector equation of a plane. The objective of curating these solutions is to promote the math potential of kids to help them accomplish their goals.

### Do I Need to Practice all Questions Provided in NCERT Solutions Class 12 Maths Three Dimensional Geometry?

Practicing all the questions included in NCERT Solutions Class 12 Maths Three Dimensional Geometry gives a step-by-step knowledge of each topic. Questions in these solutions are well-placed at a gradually increasing level to help students comfortably adjust to the rising complexity. Kids can quickly grasp the core fundamentals of three-dimensional geometry by solving all these sums. Learning with these solutions is also beneficial in preparing for various competitive exams and higher-level studies.

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

Class 12 Maths Chapter 11 Three Dimensional Geometry has 36 questions in 3 exercises. Practicing these questions ensures an accurate and clear understanding of this topic. Students can also plan systematic learning of the various topics covered in this chapter. The lucid format of these solutions is efficient in acquiring a sound knowledge of the concepts and methods with ease.

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

NCERT Solutions Class 12 Maths Chapter 11 mainly focuses on delivering the knowledge of different methods used for finding the direction of cosines and ratios. Apart from this these resources also help in learning the Cartesian and vector equation of a line. These concepts are covered in detail with the help of examples and illustrations. The format of these solutions is focused on imparting a clear understanding of fundamentals.

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

NCERT Solutions for Class 12 Maths Chapter 11 Three Dimensional Geometry is a competent learning resource that offers complete guidance and practice of core concepts for higher-level studies. These solutions cover the whole syllabus of CBSE Class 12 Maths Chapter 11 to provide comprehensive learning of all topics. The format of these solutions is pretty convenient to clear doubts and learn complex concepts. With the thorough practice of these resources, students can ensure good results in their annual maths exam.

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