NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1 Three Dimensional Geometry  PDF Download
JEE Mains & AdvancedNCERT solutions for class 12 maths chapter 11 exercise 11.1 Three Dimensional Geometry is about establishing the direction cosines and direction ratios of a line. This exercise also explains the direction cosines of a line passing through two points. These concepts of three dimensional geometry may appear difficult for students but the thorough practice of questions included in ex 11.1 class 12 maths, you will be able to eliminate the errors and doubts. Our subject matter experts of eSaral have prepared these solutions for ex 11.1 class 12 maths chapter 11 with indepth knowledge of concepts that will help you to score good marks in exams.
There are 5 questions in ex 11.1 class 12 maths ch 11. By solving questions of ex 11.1 with the help of NCERT solutions will increase your problem solving skills to find the direction cosines and direction ratio of a line. NCERT solutions for ex 11.1 class 12 maths are also provided in PDF format at eSaral. These solution PDFs can be downloaded for free from the official website of eSaral and practice all the questions offline at your own pace. These solution PDFs will establish the core foundation for three dimensional geometry with the help of simple and precise solutions of questions and examples provided in ex 11.1 class 12 maths. Download the NCERT solution PDF form the link mentioned below.
Topics Covered in Exercise 11.1 Class 12 Mathematics Questions
Class 12 maths chapter 11 exercise 11.1 NCERT solutions is based on direction cosines and direction ratios of a line, direction cosines of a line passing through two points.
1. 
Direction Cosines and Direction Ratios of a Line 
Direction cosines of a line passing through two points 

Direction Cosines and Direction Ratios of a Line
In the previous chapter, You have learnt that if a directed line L passing through the origin makes angles α, β and γ with x, y and zaxes, respectively, called direction angles, then cosine of these angles, namely, cos α, cos β and cos γ are called direction cosines of the directed line L.
If we reverse the direction of L, then the direction angles are replaced by their supplements, i.e., ,  and  . Thus, the signs of the direction cosines are reversed.
Remember that a given line in space can be extended in two opposite directions and that is why it has two sets of direction cosines. In order to have a unique set of direction cosines for a given line in space, you must take the given line as a directed line. The unique direction cosines are denoted by l, m and n.
If l, m, n are the direction cosines of a line, then l2 + m2 + n2 = 1.
Direction Ratio  Any three numbers which are proportional to the direction cosines of a line are called the direction ratios of the line. If l, m, n are direction cosines and a, b, c are direction ratios of a line, then a = λl, b=λm and c = λn, for any nonzero λ ∈ R.
If l, m, n are the direction cosines and a, b, c are the direction ratios of a line then
$l=\frac{a}{\sqrt{a^2+b^2+c^2}} ; m=\frac{b}{\sqrt{a^2+b^2+c^2}} ; n=\frac{c}{\sqrt{a^2+b^2+c^2}}$

Direction cosines of a line passing through two points
Direction cosines of a line joining two points P(x1 , y1 , z1 ) and Q(x2 , y2 , z2 ) are
$\frac{x_2x_1}{\mathrm{PQ}}, \frac{y_2y_1}{\mathrm{PQ}}, \frac{z_2z_1}{\mathrm{PQ}}$
where PQ = $\sqrt{\left(x_2x_1\right)^2+\left(y_2y_1\right)^2+\left(z_2z_1\right)^2}$
Tips for Solving Exercise 11.1 Class 12 Chapter 11 Three Dimensional Geometry
To understand the concepts and solve the questions of ex 11.1 class 12 maths, our experts of eSaral have combined some tips that you can check below.

With the thorough practice of all the questions presented in ex 11.1 class 12 maths chapter 11 NCERT solutions, you will comprehend the core concepts without being confused.

Reading and learning all the topics in NCERT solutions for ex 11.1 class 12 maths are the easiest ways to solve the questions.

It is also important to follow an illustrative format of NCERT solutions that will provide you a deep understanding of complex concepts.
Importance of Solving Ex 11.1 Class 12 Maths Chapter 11 Three Dimensional Geometry
Here, we have combined some important benefits for solving questions of ex 11.1 class 12 maths chapter 11 Three Dimensional Geometry.

Ex 11.1 class 12 maths ch 11 explains the concepts of direction cosines and ratio in simple and precise language for better understanding of questions asked in board exams.

Each question of ex 11.1 class 12 maths is solved in stepwise format and straightforward manner by expert teachers of eSaral that helps in preparing for exams.

NCERT solutions for class 12 maths chapter 11 exercise 11.1 are presented in an accurate and concise manner to grasp the concepts quickly and easily.

The NCERT solution PDF also provides the detailed answers for ex 11.1 which can be crosschecked by you anytime.
Frequently Asked Questions
Question 1. What is the direction cosines of a line?
Answer 1. Direction cosines of a line are the cosines of the angles made by the line with the positive directions of the coordinate axes.
Question 2. What are the direction ratios of a line?
Answer 2. Direction ratios of a line are the numbers which are proportional to the direction cosines of a line.
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