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# NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.2 Three Dimensional Geometry - PDF Download

NCERT solutions for class 12 maths chapter 11 exercise 11.2 Three Dimensional Geometry has questions related to vector and cartesian equations of a line in space, angle between two lines and shortest distance between two skew lines and parallel lines. These topics can be solved by continuous practice of questions included in this exercise. eSaral’s academic team of mathematics have developed the solutions of these topics in detailed format for a deep and clear understanding of concepts that will help you to solve questions without any doubt.

Class 12 maths chapter 11 exercise 11.2 NCERT solutions consists of a total of 15 questions related to solving the equation of a line in space, finding the angle between two lines and proving the statements. Ex 11.2 class 12 maths chapter 11 is provided here in PDF format which gives step by step solutions for all questions. You can download these PDFs for free of cost from the official website of eSaral and start your preparation for board exams.

## Topics Covered in Exercise 11.2 Class 12 Mathematics Questions

NCERT solutions for ex 11.2 class 12 maths chapter 11 covers topics based on the equation of a line in space,angle between two lines, shortest distance between two lines, distance between two skew lines and distance between parallel lines. These topics are elaborated in detail by the subject experts of eSaral.

 1 Equation of a Line in Space Equation of a line through a given point and parallel to \vec{a} given vector \vec{b} 2 Angle between Two Lines 3 Shortest Distance between Two Lines Distance between two skew lines Distance between parallel lines
1. Equation of a Line in Space

You have studied the equation of lines in two dimensions in previous class, we shall now study the vector and cartesian equations of a line in space.

A line is uniquely determined if

(i) it passes through a given point and has given direction, or

(ii) it passes through two given points.

• Equation of a line through a given point and parallel to $\vec{a}$ given vector $\vec{b}$

Suppose the line passes through a point P(x1, y1, z1) and it is parallel to a vector given as   $\vec{b}=a \hat{i}+b \hat{j}+c \hat{k}$

Then the cartesian equation of the line

$\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}$

1. Angle between Two Lines

Angle between two lines with directed line segments a1 , b1 , c1 and a2 , b2 , c2 , respectively is given by:

$\cos \theta=\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right|$

Two lines with direction ratios a1 , b1 , c1 and a2 , b2 , c2 are

(i) perpendicular i.e. if θ = 90°

a1 a2 + b1 b2 + c1 c2 = 0

(ii) parallel i.e. if θ = 0

$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

1. Shortest Distance between Two Lines

If two lines in space intersect at a point, then the shortest distance between them is zero. Also, if two lines in space are parallel, then the shortest distance between them will be the perpendicular distance.

In a space, there are lines which are neither intersecting nor parallel. In fact, such pairs of lines are non coplanar and are called skew lines.

By the shortest distance between two lines we mean the join of a point in one line with one point on the other line so that the length of the segment so obtained is the smallest. For skew lines, the line of the shortest distance will be perpendicular to both the lines.

• Distance between two skew lines

Shortest distance between two skew lines is the line segment perpendicular to both the lines.

Shortest distance between = $\vec{r}=\vec{a}_1+\lambda \vec{b}_1$ and $\vec{r}=\vec{a}_2+\mu \vec{b}_2$ is

$\left|\frac{\left(\vec{b}_1 \times \vec{b}_2\right) \cdot\left(\vec{a}_2-\vec{a}_1\right)}{\left|\vec{b}_1 \times \vec{b}_2\right|}\right|$

• Distance between parallel lines

Distance between parallel lines $\vec{r}=\vec{a}_1+\lambda \vec{b}$ and $\vec{r}=\vec{a}_2+\mu \vec{b}$ is

$\left|\frac{\vec{b} \times\left(\vec{a}_2-\vec{a}_1\right)}{|\vec{b}|}\right|$

## Tips for Solving Exercise 11.2 Class 12 Chapter 11 Three Dimensional Geometry

You can solve questions of ex 11.2 class 12 maths chapter 11 by following the tips provided by expert teachers of eSaral.

1. There are properties such as skew lines and angle between skew lines are explained deeply with easy and precise methods that you must study before solving questions.

2. By solving questions in NCERT solutions will give you an in-depth understanding of concepts.

3. You must learn the definition, terms, formulas and properties of topics associated with ex 11.2 to solve questions without any doubt.

## Importance of Solving Ex 11.2 Class 12 Maths Chapter 11 Three Dimensional Geometry

There are a lot of advantages of solving ex 11.2 class 12 maths chapter 11 Three Dimensional Geometry. Here, our experts have provided some of the benefits for your convenience.

1. Solving questions of ex 11.2 class 12 maths ch 11 NCERT solutions will help you to score good marks in exams.

2. All the properties and formulas are explained with simple language so that you can understand the concepts.

3. Practicing examples and questions of ex 11.2 class 12 maths will enhance your time management skills.