## Three Dimensional Geometry Examples

**Question 1: If a line makes angles 90°, 135°, 45° with the x, y and z-axes respectively, find its direction cosines.**

Solution:

Let the direction cosines of the line be l, m and n.

Here let = 90°, = 135° and = 45°

As we know,

l = cos , m = cos and n = cos

Thus, direction cosines are:

Therefore, the points A, B and C are collinear.

## 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.

## Important Questions On 12th Maths Chapter 11

##### What is the relation between the direction cosines of a line?

If l, m, n are the direction cosines of a line, then l² + m² + n² = 1

##### What is meant by Direction ratios of a line?

Direction ratios of a line are the numbers which are proportional to the direction cosines of the line.

##### What are Skew lines in 3-D geometry?

Skew lines are lines in the space which are neither parallel nor intersecting. They lie in the different planes.

##### What is the shortest distance between two skew lines?

The shortest distance between two skew lines is the length of the line segment perpendicular to both the lines.

##### Find the Equation of a plane which is at a distance p from the origin with direction cosines of the normal to the plane as l, m, n.

Equation of a plane which is at a distance p from the origin with direction cosines of the normal to the plane as l, m, n is lx + my + nz = p.

**Also Check: What Does G Stand For In Physics **

## Cbse Class 12 Mathematics Three Dimensional Geometry Mcqs Set A

CBSE Class 12 Mathematics Three Dimensional Geometry MCQs Set A with answers available in Pdf for free download. The MCQ Questions for Class 12 Three Dimensional Geometry with answers have been prepared as per the latest syllabus, NCERT books and examination pattern suggested in Standard 12 by CBSE, NCERT and KVS. Multiple Choice Questions are an important part of exams for Grade 12 Three Dimensional Geometry and if practiced properly can help you to get higher marks. Refer to more Chapter-wise MCQs for NCERT Class 12 Three Dimensional Geometry and also download more latest study material for all subjects

## 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 l2 + m2 + n2 = 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 l1, m1, n1 and l2, m2, n2 are the direction cosines of two lines and is the acute angle between the two lines then cos = |l1l2 + m1m2 + n1n2|

**Read Also: Who Is The Biological Father Of Paris Jackson **

## Important Questions & Answers For Class 12 Maths Chapter 11 Three Dimensional Geometry

**Q. 1: Find the direction cosines of the line passing through the two points and .**

**Solution:**

We know that the direction cosines of the line passing through two points P and Q are given by

Using the distance formula,

From the given,

P = and Q =

Hence, the direction cosines of the line joining the given two points are .

**Q. 2: Show that the points A , B and C are collinear.**

**Solution:**

We know that the direction ratios of the line passing through two points P and Q are given by:

x2 x1, y2 y1, z2 z1 or x1 x2, y1 y2, z1 z2

Given points are A , B and C .

Direction ratios of the line joining A and B are:

1 2, 2 3, 3 + 4

i.e. 1, 5, 7.

The direction ratios of the line joining B and C are:

3 1, 8 + 2, 11 3

i.e., 2, 10, 14.

From the above, it is clear that direction ratios of AB and BC are proportional.

That means AB is parallel to BC. But point B is common to both AB and BC.

Hence, A, B, C are collinear points.

**Q. 3: If a line makes angles 90°, 135°, 45° with the x, y and z-axes respectively, find its direction cosines.**

**Solution:**

Let the direction cosines of the line be l, m, and n.

l = cos 90° = 0

m = cos 135° = -1/2

n = cos 45° = 1/2

Hence, the direction cosines of the line are 0, -1/2, and 1/2.

**Q. 4: Find the angle between the pair of lines given by **

**Solution:**

Let be the angle between the given pair of lines.

**Q. 5: Find the angle between the pair of lines given below.**

**/3 = /5 = /4**

**/1 = /1 = /2**

**Solution:**

/3 = /5 = /4

/1 = /1 = /2

a1 = 3, b1 = 5, c1 = 4

## Three Dimensional Geometry Important Extra Questions Short Answer Type

Question 1.Find the acute angle between the lines whose direction-ratios are:< 1,1,2 > and < -3, -4,1 > .

Find the angle between the following pair of lines:and\ and \and check whether the lines are parallel or perpendicular. Solution:The given lines can be rewritten as :\ .. \ ..Here < 2,7, 3 > and < -1,2,4 > are direction- ratios of lines and respectively.Hence, the given lines aife perpendicular.

Question 3.Find the vector equation of the line joining and and show that it is perpendicular to the z-axis. Solution:Vector equation of the line passing through andis \\)where \ and \ \+\lambda\) Equation of z-axis is \ Since \ \cdot \hat=0\) = 0, Line is perpendicular to z-axis.

Question 4.Find the vector equation of the plane, which is \ at a distance ofunits from the origin and its normal vector from the origin is \ . Also, find its cartesian form. Solution:

Question 5.Find the direction-cosines of the unit vector perpendicular to the plane \\) +1 = 0 through the origin. Solution:The given plane is \\) + 1 = 0\\) = 1 Now \\Dividing by 7,\=\frac\)which is the equation of the plane in the form \Thus, \which is the unit vector perpendicular to the plane through the origin.Hence, the direction-cosines of \ are \

Question 6.Find the acute angle between the lines\ and \Solution:Vector in the direction of first line\ ,\\)

Vector in the direction of second line\ ,\ , the angle between two given lines is given by:

**Don’t Miss: Eoc Fsa Warm Ups Algebra 1 Answers **

## 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.

## Practice Questions For Class 12 Maths Chapter 11 Three Dimensional Geometry

MATHS Related Links |

**Recommended Reading: Elimination Method Steps **

## Three Dimensional Geometry For Class 12

**Three Dimensional Geometry for class 12** covers important topics such as direction cosine and direction ratios of a line joining two points. Also, we will discuss here, the equation of lines and planes in space under different condition, the angle between line and plane, between two lines etc. You need to practice the questions to understand the topic better and based on the formulas as well.

To understand the different types of shapes and figures, this topic has been introduced in Maths. In the real world, almost all the objects are in a three-dimensional shape. For example, there are many objects at home such as a table, chair, bed, kitchen utensils, etc. which have 3D geometry. In our primary classes, we have learned the basics of three-dimension geometry, but in the 12th standard, we will learn the advanced version of it.

## Cbse Class 12 Maths Chapter

Free PDF download of Important Questions for CBSE Class 12 Maths Chapter 11 Three Dimensional Geometry prepared by expert Maths teachers from latest edition of CBSE books, On CoolGyan.Org to score more marks in CBSE board examination. You can also Download __Maths Revision Notes Class 12__ to help you to revise complete Syllabus and score more marks in your examinations.

**You May Like: Exponential Growth And Decay Common Core Algebra 1 Homework Answers **

## Direction Cosine And Direction Ratios Of A Line

Consider a line L passing through origin makes an angle of , , with x, y, and z-axes respectively, then the cosine of these angles is the direction cosine of the directed line L.

Any three numbers which are proportional to the direction cosines of a line are called the direction ratios of the line. Consider the direction cosine of line L be l, m, n and direction ratio a, b, c then a = l, b = m, c = n, for non-zero R.

Then the direction cosine are:

**Note:** If the given line in space does not pass through the origin, then, in order to find its direction cosines, we draw a line through the origin and parallel to the given line. Now take one of the directed lines from the origin and find its direction cosines as two parallel line have same set of direction cosines.

## Faqs Related To Ncert Solutions For Class 12 Maths Chapter 11 Pdf

**Q. Where can I get NCERT Solutions for Class 12 Maths PDF to download for all chapters?**A. Follow the link **NCERT Solutions for Class 12 Maths** to download for all chapters PDF on Embibe.

**Q. What is the name of CBSE NCERT Solutions for Class 12 Maths Chapter 11?**A. Class 12 Maths Chapter 11 name is Three Dimensional Geometry.

**Q. How many exercises are there in NCERT Solutions for Class 12 Maths Chapter 11 PDF**?A. There is a total of 10 exercises with subtopics. in Class 12 Maths Chapter 11.

**Q. What is the formula for the angle between two lines of Class 12 Maths?**A. If one of the line is parallel to y-axis then the angle between two straight lines is given by tan = ±1/m where m is the slope of the other straight line. If the two lines are a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, then the formula becomes tan = |/|

**Q. How do you calculate a plane class 12 Maths?**A. We get the point-normal equation A+B+C = 0. for a plane. for the equation of a plane having normal n=A,B,C. Here D=nb=Aa+Bb+Cc.

Embibe provides **CBSE study material** that covers the whole CBSE Class 12 syllabus for Maths. You can also solve **Maths practice questions** for every chapter in the CBSE Class 12 syllabus for Maths that will also help you in your preparation of JEE as well.

*We hope this article has been helpful to you. If you have any queries/ doubts, leave them in the comment section below and we will get back to you at the earliest.*

**Also Check: Kendall Hunt Geometry Answer Key **

## Xam Idea Books Solutions Class 12

Xam idea Class 12 Mathematics Solutions is a unique learning experience. Every book is divided into two parts such as Part A and Part B. Part A include the Basic Concepts of the Chapters, Higher Order Thinking Skills questions, important NCERT questions and important questions of Previous Years examinations. Part B covers the latest solved CBSE Sample Papers, Model Question Papers for practice and Previous Years solved Examination Papers.

## Maths Mcqs For Class 12 With Answers Chapter 11 Three Dimensional Geometry

Free PDF Download of CBSE Maths Multiple Choice Questions for Class 12 with Answers Chapter 11 Three Dimensional Geometry. Maths MCQs for Class 12 Chapter Wise with Answers PDF Download was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 12 Maths Three Dimensional Geometry MCQs Pdf with Answers to know their preparation level.

**Recommended Reading: Eoc Fsa Warm Ups Algebra 1 Answers **

## Xam Idea Class 12 Mathematics Solutions

We provide you with one of the best and reliable Xam Idea Class 12 Mathematics Solutions. The Class 12 Mathematics Solutions are developed by the SelfStudys expert and skilled team of teachers who have huge experience, covering the entire syllabus of the exam. Scoring high marks in all subject will become easy if you adopt the correct approach and make your fundamentals strong in the subject. So, to help you in your preparation of the exam, our Xam Idea Class 12 Mathematics Solutions are prepared in such a way covering all the questions-answers in the great-structured format.

The Class 12 Mathematics Solutions of Xam Idea holds high regard among students of CBSE Class 12. The reference Class 12 Mathematics Solutions are in accordance with the latest CBSE exam pattern. Every chapter has been explored fully for the advantage of both students and teachers. The Xam Idea Class 12 Mathematics Solutions are provided here so that students can read the concepts of the chapter more efficiently.

## Direction Cosines Of A Line Passing Through Two Points

The direction cosines of the line segment joining the points P and Q are:

**x2-x1/PQ, y2-y1/PQ, z2-z1/PQ**

where PQ =

- The shortest distance between two skew lines is the line segment perpendicular to both lines.
- If l, m, n are the direction cosines, and a, b, c are the direction ratios of the line, then the direction cosines are given by:

The direction cosines of two lines are l1 , m1 , n1 and l2 , m2 , n2 and is the acute angle between the two lines then cos = | l1 l 2 + m1 m2 + n1 n2|.

**Also Check: What Are Dyes In Chemistry **

## Features Of Xam Idea Class 12 Mathematics Solutions

- The Xam Idea Class 12 Solutions cover the complete syllabus of the exam.
- The skilled and well- experienced subject teachers who hold huge experience has prepared these Class 12 Mathematics Solutions.
- Clear your doubts and get step by step answers to the questions asked in the Xam Idea Class 12 textbook.
- You can access these Class 12 Mathematics Solutions at free of cost.
- Getting good marks in the exam will become easy for you if you plan your preparation from the Xam Idea Class 12 Mathematics Solutions.
- The easy to understand language and well-structured format is sure to help you in your preparation for the examination.

## Cbse Class 12th Maths Most Important Questions Chapter Wise Pdf Download

According to my, you should go with the previous year questions of maths 12th CBSE, before going through important questions of mathematics chapter-wise. In this article, I have covered the most important questions for 12th CBSE Maths, which all have a great chance of coming to your board exam this year.

As you know that mathematics is the only subject in which you can score 100 out of 100 marks if you try, and once you develop an interest in it, then things will get so much interesting and fun.

Preparing mathematics is easier as compared to other subjects like physics and chemistry, because mathematics is totally based on numerical problems, and a number of questions in mathematics repeat every year. One can easily make an idea about the upcoming questions by solving the CBSE sample papers for class 12 Mathematics.

**Don’t Miss: What Is The Molecular Geometry Of Ccl4 **