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Edited By Ramraj Saini | Updated on Dec 04, 2023 09:10 AM IST | #CBSE Class 12th

**NCERT Solutions for Exercise 11.2 Class 12 Maths Chapter 11 Three Dimensional Geometry **are discussed here. These NCERT solutions are created by subject matter expert at Careers360 considering the **latest syllabus and pattern of CBSE 2023-24. **NCERT solutions for exercise 11.2 Class 12 Maths chapter 11 is about the lines in space and the equation of these lines in cartesian and vector form. NCERT solutions for Class 12 Maths chapter 11 exercise 11.2 also covers the topic of the distance between lines. The topics covered in the exercise 11.2 Class 12 Maths are very important if the CBSE Class 12 Maths Previous Paper is considered. The main focus of solving Class 12 Maths chapter 11 exercise 11.2 should be to check whether the concepts are grasped or not.

**12th class Maths exercise 11.2 **answers** **are designed as per the students demand covering comprehensive, step by step solutions of every problem. Practice these questions and answers to command the concepts, boost confidence and in depth understanding of concepts. Students can find all exercise together using the link provided below.

- Three Dimensional Geometry 11.1
- Three Dimensional Geometry 11.3
- Three Dimensional Geometry Miscellaneous Exercise

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are mutually perpendicular.

** Answer: **

GIven direction cosines of the three lines;

And we know that two lines with direction cosines and are perpendicular to each other, if

Hence we will check each pair of lines:

** Lines ; **

the lines are perpendicular.

** Lines ; **

the lines are perpendicular.

** Lines ; **

the lines are perpendicular.

Thus, we have all lines are mutually perpendicular to each other.

** Answer: **

We have given points where the line is passing through it;

Consider the line joining the points (1, – 1, 2) and (3, 4, – 2) is AB and line joining the points (0, 3, 2) and (3, 5, 6).is CD.

So, we will find the direction ratios of the lines AB and CD;

Direction ratios of AB are

or

Direction ratios of CD are

or .

Now, lines AB and CD will be perpendicular to each other if

** Therefore, AB and CD are perpendicular to each other. **

** Answer: **

We have given points where the line is passing through it;

Consider the line joining the points (4, 7, 8) and (2, 3, 4) is AB and line joining the points (– 1, – 2, 1) and (1, 2, 5)..is CD.

So, we will find the direction ratios of the lines AB and CD;

Direction ratios of AB are

or

Direction ratios of CD are

or .

Now, lines AB and CD will be parallel to each other if

Therefore we have now;

** Hence we can say that AB is parallel to CD. **

** Answer: **

It is given that the line is passing through A (1, 2, 3) and is parallel to the vector

We can easily find the equation of the line which passes through the point A and is parallel to the vector by the known relation;

, where is a constant.

So, we have now,

** Thus the required equation of the line. **

** Answer: **

Given that the line is passing through the point with position vector and is in the direction of the line .

And we know the equation of the line which passes through the point with the position vector and parallel to the vector is given by the equation,

** So, this is the required equation of the line in the vector form. **

Eliminating , from the above equation we obtain the equation in the Cartesian form :

** Hence this is the required equation of the line in Cartesian form. **

** Answer: **

Given a line which passes through the point (– 2, 4, – 5) and is parallel to the line given by the ;

The direction ratios of the line, are ** 3,5 and 6 ** .

So, the required line is parallel to the above line.

Therefore we can take direction ratios of the required line as ** 3k ** , ** 5k ** , and ** 6k ** , where k is a non-zero constant.

And we know that the equation of line passing through the point and with direction ratios a, b, c is written by: .

Therefore we have the equation of the required line:

or

** The required line equation. **

** Question:7 ** The cartesian equation of a line is . Write its vector form .

** Answer: **

Given the Cartesian equation of the line;

Here the given line is passing through the point .

So, we can write the position vector of this point as;

And the direction ratios of the line are ** 3 ** , ** 7 ** , and ** 2. **

This implies that the given line is in the direction of the vector, .

Now, we can easily find the required equation of line:

As we know that the line passing through the position vector and in the direction of the vector is given by the relation,

So, we get the equation.

** This is the required equation of the line in the vector form. **

** Answer: **

GIven that the line is passing through the and

Thus the required line passes through the origin.

its position vector is given by,

So, the direction ratios of the line through and are,

The line is parallel to the vector given by the equation,

Therefore the equation of the line passing through the point with position vector and parallel to is given by;

Now, the equation of the line through the point and the direction ratios a, b, c is given by;

Therefore the equation of the required line in the Cartesian form will be;

OR

** Answer: **

Let the line passing through the points and is AB;

Then as AB passes through through A so, we can write its position vector as;

Then direction ratios of PQ are given by,

Therefore the equation of the vector in the direction of AB is given by,

We have then the equation of line AB in vector form is given by,

So, the equation of AB in Cartesian form is;

or

** Question:10 ** Find the angle between the following pairs of lines:

(i) and

** Answer: **

To find the angle A between the pair of lines we have the formula;

We have two lines :

and

The given lines are parallel to the vectors ;

where and respectively,

Then we have

and

Therefore we have;

or

** Question:10 ** Find the angle between the following pairs of lines:

(ii) and

** Answer: **

To find the angle A between the pair of lines we have the formula;

We have two lines :

** Question:11 ** Find the angle between the following pair of lines:

(i) and

** Answer: **

Given lines are;

and

So, we two vectors which are parallel to the pair of above lines respectively.

and

To find the angle A between the pair of lines we have the formula;

Then we have

and

Therefore we have;

or

** Question:11 ** Find the angle between the following pair of lines:

(ii) and

** Answer: **

Given lines are;

and

So, we two vectors which are parallel to the pair of above lines respectively.

and

To find the angle A between the pair of lines we have the formula;

Then we have

and

Therefore we have;

or

** Question:12 ** Find the values of p so that the lines and are at right angles.

** Answer: **

First we have to write the given equation of lines in the standard form;

and

Then we have the direction ratios of the above lines as;

and respectively..

Two lines with direction ratios and are perpendicular to each other if,

Thus, the value of p is .

** Question:13 ** Show that the lines and are perpendicular to each other.

** Answer: **

First, we have to write the given equation of lines in the standard form;

and

Then we have the direction ratios of the above lines as;

and respectively..

Two lines with direction ratios and are perpendicular to each other if,

Therefore the two lines are perpendicular to each other.

** Question:14 ** Find the shortest distance between the lines

and

** Answer: **

So given equation of lines;

and in the vector form.

Now, we can find the shortest distance between the lines and , is given by the formula,

Now comparing the values from the equation, we obtain

Then calculating

So, substituting the values now in the formula above we get;

** Therefore, the shortest distance between the two lines is units. **

** Question:15 ** Find the shortest distance between the lines

and

** Answer: **

We have given two lines:

and

Calculating the shortest distance between the two lines,

and

by the formula

Now, comparing the given equations, we obtain

Then calculating determinant

Now calculating the denominator,

So, we will substitute all the values in the formula above to obtain,

Since distance is always non-negative, the distance between the given lines is

units.

** Question:16 ** Find the shortest distance between the lines whose vector equations are and

** Answer: **

Given two equations of line

in the vector form.

So, we will apply the distance formula for knowing the distance between two lines and

After comparing the given equations, we obtain

Then calculating the determinant value numerator.

That implies,

Now, after substituting the value in the above formula we get,

Therefore, is the shortest distance between the two given lines.

** Question:17 ** Find the shortest distance between the lines whose vector equations are

and

** Answer: **

Given two equations of the line

in the vector form.

So, we will apply the distance formula for knowing the distance between two lines and

After comparing the given equations, we obtain

Then calculating the determinant value numerator.

That implies,

Now, after substituting the value in the above formula we get,

Therefore, units are the shortest distance between the two given lines.

Ten solved questions are given prior to the exercise 11.2 Class 12 Maths. And 17 questions are covered in Class 12 Maths chapter 11 exercise 11.2. Broadly speaking Class 12th Maths chapter 11 exercise 11.2 covers questions related to the equation of a line parallel to a given vector and that passes through a given point, the line passing through two given points, the angle between lines, the smallest distance between two lines and distance between the skew lines and parallel lines.

**Also Read| **Three Dimensional Geometry Class 12th Notes

**Benefits of NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.2**

- Solving the given NCERT examples for topic 11.3 and exercise 11.2 Class 12 Maths help to score well in the exam
- Question of similar type in Class 12th Maths chapter 11 exercise 11.2 can be expected for CBSE board exam.

JEE Main Highest Scoring Chapters & Topics

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Download EBook**Comprehensive Coverage:**The solutions encompass all the topics covered in ex 11.2 class 12, ensuring a thorough understanding of the concepts.**Step-by-Step Solutions:**In this class 12 maths ex 11.2, each problem is solved systematically, providing a stepwise approach to aid in better comprehension for students.**Accuracy and Clarity:**Solutions for class 12 ex 11.2 are presented accurately and concisely, using simple language to help students grasp the concepts easily.**Conceptual Clarity:**In this 12th class maths exercise 11.2 answers, emphasis is placed on conceptual clarity, providing explanations that assist students in understanding the underlying principles behind each problem.**Inclusive Approach:**Solutions for ex 11.2 class 12 cater to different learning styles and abilities, ensuring that students of various levels can grasp the concepts effectively.**Relevance to Curriculum:**The solutions for class 12 maths ex 11.2 align closely with the NCERT curriculum, ensuring that students are prepared in line with the prescribed syllabus.

- NCERT Solutions Class 12 Chemistry
- NCERT Solutions for Class 12 Physics
- NCERT Solutions for Class 12 Biology
- NCERT Solutions for Class 12 Mathematics

1. What is the first question of exercise 11.2 Class 12 Maths about?

The question is to show three lines with given direction cosines are perpendicular

2. Write down the condition for lines with direction cosines l1, m1, n1 and l2, m2 and n2 to be perpendicular?

l1l^{2}+m^{1}m^{2}+n^{1}n^{2}=0

3. How many questions are solved in the NCERT solutions for Class 12 Maths chapter 11 exercise 11.2?

17 questions are solved in the exercise 11.2 Class 12 Maths

4. Which questions depict the concept of the angle between lines?

Question 10 to 13 of Class 12th Maths chapter 11 exercise 11.2

5. Give the topic covered in questions 14 to 17?

Questions 14 to 17 covers the concepts of the shortest distance between two lines.

6. What is the topic discussed after exercise 11.2?

The topic plane is discussed after exercise 11.2

7. What are the main topics covered in Class 12 Maths exercise 11.2?

The main topics covered are equations of lines in three dimensions, angles between lines and least distance between lines.

8. How many exercises are discussed in chapter three dimensional geometry?

4 exercises including the miscellaneous.

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6 minHave a question related to CBSE Class 12th ?

You can use them people also used problem

Hi,

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hello mahima,

If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.

hope this helps.

Hello Akash,

If you are looking for important questions of class 12th then I would like to suggest you to go with previous year questions of that particular board. You can go with last 5-10 years of PYQs so and after going through all the questions you will have a clear idea about the type and level of questions that are being asked and it will help you to boost your class 12th board preparation.

You can get the Previous Year Questions (PYQs) on the official website of the respective board.

I hope this answer helps you. If you have more queries then feel free to share your questions with us we will be happy to assist you.

**
Thank you and wishing you all the best for your bright future.
**

Hello student,

**
If you are planning to appear again for class 12th board exam with PCMB as a private candidate here is the right information you need:
**

- No school admission needed! Register directly with CBSE. (But if you want to attend the school then you can take admission in any private school of your choice but it will be waste of money)
- You have to appear for the 2025 12th board exams.
- Registration for class 12th board exam starts around September 2024 (check CBSE website for exact dates).
- Aim to register before late October to avoid extra fees.
- Schools might not offer classes for private students, so focus on self-study or coaching.

**
Remember
**
, these are tentative dates based on last year. Keep an eye on the CBSE website ( https://www.cbse.gov.in/ ) for the accurate and official announcement.

I hope this answer helps you. If you have more queries then feel free to share your questions with us, we will be happy to help you.

**
Good luck with your studies!
**

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