Introduction To Three Dimensional Geometry Class 11th Notes - Free NCERT Class 11 Maths Chapter 12 notes - Download PDF

Have you ever wondered how large skyscrapers are designed or how the GPS tells us about our exact locations every time? To find the answer, we have to study three-dimensional geometry, where we go beyond our understanding of two-dimensional geometry to the X, Y, and Z coordinate axes. From NCERT Class 11 Maths, the chapter Introduction to Three-Dimensional Geometry contains the concepts of Coordinate Axes and Coordinate Planes in Three-Dimensional Space, Coordinates of a Point in Space, Distance between Two Points and its formula, etc. These concepts will help the students grasp more advanced geometry concepts easily and enhance their problem-solving ability in real-world applications.

This article on NCERT notes Class 11 Maths Chapter 11 Introduction to Three-Dimensional Geometry offers well-structured NCERT notes to help the students grasp the concepts of three-dimensional geometry easily. Students who want to revise the key topics of three-dimensional geometry quickly will find this article very useful. It will also boost the exam preparation of the students several times. These notes of NCERT Class 11 Maths Chapter 11 Introduction to Three-Dimensional Geometry are made by the Subject Matter Experts according to the latest CBSE syllabus, ensuring that students can grasp the basic concepts effectively. NCERT solutions for class 11 maths and NCERT solutions for other subjects and classes can be downloaded from the NCERT Solutions.

NCERT Class 11 Math Chapter 11 Notes

Distance Formula

Suppose that there are two points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ then the distance formula is given as

$d=\sqrt{{(x_1-x_2)}^2+{(y_1-y_2)}^2+{(z_1-z_2)}^2}$

Section Formulas

Suppose that there are two points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ are divided into $m:n$.

The section formula is given as:

$(x^{\prime },y^{\prime },z^{\prime })=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n},\frac{m{z}_2+n{z}_1}{m+n})$

Systems Of Planes

An equation of a plane contains three independent constants; as such, a plane is uniquely determined by three independent conditions. If we consider a plane satisfying just two given conditions, its equation will contain one arbitrary constant. If we consider a plane satisfying one given condition, its equation will contain two arbitrary constants.

We give below the equations of some well-known systems of planes:

The equation $ax+by+cz+k=0$ represents a system of planes parallel to the plane $ax+by+cz+d=0$, where $d$ and $k$ are parameters.

The equation $ax+by+cz+k=0$ represents a system of planes perpendicular to the line $\frac{x}{a}=\frac{y}{b}=\frac{z}{c}$.

The equation $a_{1}x+b_{1}y+c_{1}z+d_1+\lambda (a_{2}x+b_{2}y+c_{2}z+d_2)=0$ is a system of planes passing through the intersection of the planes $a_{1}x+b_{1}y+c_{1}z+d_1=0$ and $a_{2}x+b_{2}y+c_{2}z+d_2=0$, where $\lambda$ is a parameter.

Angle Between Two Planes

Let there be two planes $a_{1}x+b_{1}y+c_{1}z+d_1=0$ and $a_{2}x+b_{2}y+c_{2}z+d_2=0$

Then the angle between is given as

$\cos \theta =\left|\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\right|$

The Symmetrical Form Of The Equation Of A Line

If a straight line passes through a given point $(x_1,y_1,z_1)$ and has direction cosines $l,m,n,$ then the coordinates of any point on it satisfy the equations

$\frac{x-x_1}{l}=\frac{y-y_1}{m}=\frac{z-z_1}{n}$

Vector Equations:

The parametric vector equation of the line through a point with position vector $a$ and parallel to the vector $b$ is:
$\overrightarrow{r}=\overrightarrow{\mathrm{a}}+\lambda \overrightarrow{b}$, where $\lambda$ is a scalar parameter.

Importance of NCERT Class 11 Maths Chapter 11 Notes

NCERT Class 11 Maths Chapter 11 Notes play a vital role in helping students grasp the core concepts of the chapter easily and effectively, so that they can remember these concepts for a long time. Some important points of these notes are:

• Effective Revision: These notes provide a detailed overview of all the important theorems and formulas, so that students can revise the chapter quickly and effectively.
• Clear Concepts: With these well-prepared notes, students can understand the basic concepts effectively. Also, these notes will help the students remember the key concepts by breaking down complex topics into simpler and easier-to-understand points.
• Time Saving: Students can look to save time by going through these notes instead of reading the whole lengthy chapter.
• Exam Ready Preparation: These notes also highlight the relevant contents for various exams, so that students can get the last-minute minute very useful guidance for exams.

NCERT Class 11 Notes Chapter Wise

Subject-Wise NCERT Notes

Given below are some subject-wise links for the NCERT Notes for class 11.

• NCERT Notes Class 11 Physics
• NCERT Notes Class 11 Chemistry
• NCERT Notes Class 11 Biology

Subject-Wise NCERT Exemplar Solutions

After finishing the textbook exercises, students can use the following links to check the NCERT exemplar solutions for a better understanding of the concepts.

• NCERT Exemplar Class 11 Solutions
• NCERT Exemplar Class 11 Maths
• NCERT Exemplar Class 11 Physics
• NCERT Exemplar Class 11 Chemistry
• NCERT Exemplar Class 11 Biology

Subject-Wise NCERT Solutions

Students can also check these well-structured, subject-wise solutions.

• NCERT Solutions for Class 11 Mathematics
• NCERT Solutions for Class 11 Chemistry
• NCERT Solutions for Class 11 Physics

• NCERT Solutions for Class 11 Biology

NCERT Books and Syllabus

Students should always analyze the latest CBSE syllabus before making a study routine. The following links will help them check the syllabus. Also, here is access to more reference books.

• NCERT Book for Class 11
• NCERT Syllabus for Class 11

Important points to note:

• NCERT problems are very important in order to perform well in the exams. Students must try to solve all the NCERT problems, including miscellaneous exercises, and if needed, refer to the NCERT Solutions for Class 11 Maths.
• Students are advised to go through the NCERT Class 11 Maths Chapter-Wise Notes before solving the questions.
• To boost your exam preparation as well as for a quick revision, these NCERT notes are very useful.

Happy learning!