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.
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.
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 Cosines | Direction Ratios |
|---|---|
| Cosines of the angles made by the line with the coordinate axes | Slopes of the line along the coordinate axes |
| Determined up to a sign | Determined up to a scalar multiple |
| Satisfy l + m + n = 1 | No 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) |
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