3D Geometry Class 12 Notes PDF Download
Are you looking for a comprehensive and concise guide to learn 3D geometry for class 12? Do you want to master the concepts and formulas of 3D geometry with ease and confidence? If yes, then you have come to the right place. In this article, we will provide you with the best notes on 3D geometry class 12 that you can download in PDF format for free.
3D geometry is a branch of mathematics that deals with the study of shapes, objects, and figures in three-dimensional space. It is an extension of the two-dimensional geometry that you have learned in class 11. 3D geometry has many applications in various fields such as engineering, architecture, design, computer graphics, astronomy, etc. It also helps you to develop your spatial reasoning and visualization skills.
3d geometry class 12 notes pdf download
Download File: https://lerexgingi.blogspot.com/?fe=2vyqra
In class 12, you will learn about the following topics in 3D geometry:
Direction cosines and direction ratios of a line
Equations of a line in space
Equations of a plane in space
In this article, we will explain each topic in detail with definitions, formulas, examples, and exercises. We will also provide you with some tips and tricks to solve the problems faster and easier. By reading these notes, you will be able to understand the concepts clearly and score well in your exams.
Direction Cosines and Direction Ratios of a Line
In this section, we will learn about the direction cosines and direction ratios of a line in 3D space. These are important concepts that help us to describe the orientation and direction of a line.
Definition and notation of direction cosines and direction ratios
Consider a line L passing through the origin O in 3D space. Let α, β, and γ be the angles that L makes with the positive directions of the x-axis, y-axis, and z-axis respectively. These angles are called the direction angles of L.
3d geometry class 12 cbse maths chapter 11 revision notes pdf
class 12 maths three dimensional geometry notes pdf free download
three dimensional geometry class 12 ncert solutions pdf download
class 12 mathematics notes for chapter 11 three dimensional geometry pdf
three dimensional geometry class 12 vedantu notes pdf download
class 12 maths three dimensional geometry formulas pdf download
three dimensional geometry class 12 important questions pdf download
class 12 maths chapter 11 three dimensional geometry ncert book pdf
three dimensional geometry class 12 selfstudys revision notes pdf
class 12 maths three dimensional geometry coordinate system pdf download
three dimensional geometry class 12 direction cosines and ratios pdf download
class 12 maths three dimensional geometry angle between two lines pdf download
three dimensional geometry class 12 equation of a line in space pdf download
class 12 maths three dimensional geometry shortest distance between two skew lines pdf download
three dimensional geometry class 12 equation of a plane in normal form pdf download
class 12 maths three dimensional geometry angle between two planes pdf download
three dimensional geometry class 12 distance of a point from a plane pdf download
class 12 maths three dimensional geometry equation of a plane passing through given points pdf download
three dimensional geometry class 12 coplanarity of two lines pdf download
class 12 maths three dimensional geometry equation of a line of intersection of two planes pdf download
three dimensional geometry class 12 family of planes pdf download
class 12 maths three dimensional geometry perpendicular distance between parallel planes pdf download
three dimensional geometry class 12 bisectors of angles between two planes pdf download
class 12 maths three dimensional geometry section formulae in space pdf download
three dimensional geometry class 12 centroid of a triangle and tetrahedron pdf download
class 12 maths three dimensional geometry solved examples pdf download
three dimensional geometry class 12 exercises with answers pdf download
class 12 maths three dimensional geometry previous year questions pdf download
three dimensional geometry class 12 sample papers with solutions pdf download
class 12 maths three dimensional geometry mock tests online pdf download
three dimensional geometry class 12 video lectures free download pdf
class 12 maths three dimensional geometry study material pdf download
three dimensional geometry class 12 summary and key points pdf download
class 12 maths three dimensional geometry mind maps and cheat sheets pdf download
three dimensional geometry class 12 cbse syllabus and exam pattern pdf download
class 12 maths three dimensional geometry tips and tricks pdf download
three dimensional geometry class 12 doubt clearing sessions online pdf download
class 12 maths three dimensional geometry extra questions for practice pdf download
three dimensional geometry class 12 assignments and worksheets pdf download
class 12 maths three dimensional geometry projects and activities pdf download
three dimensional geometry class 12 reference books and guides pdf download
class 12 maths three dimensional geometry online courses and coaching classes pdf download
three dimensional geometry class 12 career options and opportunities after math stream pdf download
The cosines of these angles are called the direction cosines of L. They are denoted by l, m, and n respectively. That is,
l = cos α
m = cos β
n = cos γ
The direction cosines satisfy the following relation:
l + m + n = 1
This is because cos α + cos β + cos γ = 1, which is a trigonometric identity.
If we reverse the direction of L, then the direction angles become π - α, π - β, and π - γ respectively. Hence, the direction cosines become -l, -m, and -n respectively. Therefore, a line has two sets of direction cosines, which are opposite in sign.
The direction cosines are also related to the slope of the line. If a, b, and c are the slopes of L along the x-axis, y-axis, and z-axis respectively, then we have:
l = a/(a + b + c)
m = b/(a + b + c)
n = c/(a + b + c)
The numbers a, b, and c are called the direction ratios of L. They are not unique, as they can be multiplied by any non-zero constant and still represent the same line. However, the direction cosines are unique, as they are normalized by dividing by the magnitude of the vector.
Relation between direction cosines and direction ratios
We can summarize the relation between direction cosines and direction ratios as follows:
Direction CosinesDirection Ratios
Cosines of the angles made by the line with the coordinate axesSlopes of the line along the coordinate axes
Determined up to a signDetermined up to a scalar multiple
Satisfy l + m + n = 1No such condition
If l, m, and n are direction cosines, then k*l, k*m, and k*n are not (unless k = 1)If a, b, and c are direction ratios, then k*a, k*b, and k*c are also direction ratios (for any non-zero k)
If l, m, and n are direction cosines, then a = l*(a + b + c) , b = m*(a + b + c) , and c = n*(a + b + c) are direction ratios If a, b, and c are direction ratios, then l = a/(a + b + c) , m = b/(a + b + c) , and n = c/(a + b + c) are direction cosines
The direction vector of the line is given by v = l*i + m*j + n*k , where i, j, and k are unit vectors along the coordinate axes The direction vector of the line is given by v = a*i + b*j + c*k , where i, j, and k are unit vectors along the coordinate axes
The magnitude of the direction vector is 1 The magnitude of the direction vector is (a+bb+ 2c^c)
[assistant](#message) 44f88ac181
Comments