# NCERT Solutions for Exercise 11.1 Class 12 Maths Chapter 11- Three Dimensional Geometry

NCERT solutions for exercise 11.1 Class 12 Maths chapter 11 focus on the topic direction ratio and direction cosines of a line. Exercise 11.1 Class 12 Maths moves around the topic 11.2. After having a thorough look at the concepts and example questions, students can solve Class 12 Maths chapter 11 exercise 11.1 for good clarification of the concepts discussed before the Class 12th Maths chapter 11 exercise 11.1. If the journey of solving exercise 11.1 Class 12 Maths stops anywhere, refer to NCERT solutions for Class 12 Maths chapter 11 exercise 11.1. Along with the first exercise of Class 12 NCERT Maths chapter 11, the following exercises are also present. Solving all this exercise gives a better image of the chapter.

• Three Dimensional Geometry 11.2

• Three Dimensional Geometry 11.3

• Three Dimensional Geometry Miscellaneous Exercise

## Three Dimensional Geometry Class 12th Chapter 11 -Exercise: 11.1

Question:1 If a line makes angles $90^{\circ}, 135^{\circ},45^{\circ}$ with the x, y and z-axes respectively, find its direction cosines.

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

So, we have

$l = \cos90^{\circ}=0$

$m = \cos135^{\circ}=-\frac{1}{\sqrt2}$

$n= \cos45^{\circ}=\frac{1}{\sqrt2}$

Therefore the direction cosines of the lines are $0,\ -\frac{1}{\sqrt2},and\ \ \frac{1}{\sqrt2}$ .

Question:2 Find the direction cosines of a line which makes equal angles with the coordinate axes.

If the line is making equal angle with the coordinate axes. Then,

Let the common angle made is $\alpha$ with each coordinate axes.

Therefore, we can write;

$l = \cos \alpha,\ m= \cos \alpha,and\ n= \cos \alpha$

And as we know the relation; $l^2+m^2+n^2 = 1$

$\Rightarrow \cos^2 \alpha +\cos^2 \alpha+\cos^2 \alpha = 1$

$\Rightarrow \cos^2 \alpha = \frac{1}{3}$

or $\cos \alpha =\pm \frac{1}{\sqrt3}$

Thus the direction cosines of the line are $\pm \frac{1}{\sqrt3},\ \pm \frac{1}{\sqrt3},and\ \pm \frac{1}{\sqrt3}$

Question:3 If a line has the direction ratios –18, 12, – 4, then what are its direction cosines ?

GIven a line has direction ratios of -18, 12, – 4 then its direction cosines are;

Line having direction ratio -18 has direction cosine:

$\frac{-18}{\sqrt{(-18)^2+(12)^2+(-4)^2}} = \frac{-18}{22} = \frac{-9}{11}$

Line having direction ratio 12 has direction cosine:

$\frac{12}{\sqrt{(-18)^2+(12)^2+(-4)^2}} = \frac{12}{22} =\frac{6}{11}$

Line having direction ratio -4 has direction cosine:

$\frac{12}{\sqrt{(-4)^2+(12)^2+(-4)^2}} = \frac{-4}{22} = \frac{-2}{11}$

Thus, the direction cosines are $\frac{-9}{11},\ \frac{6}{11},\ \frac{-2}{11}$ .

Question:4 Show that the points (2, 3, 4), (– 1, – 2, 1), (5, 8, 7) are collinear.

We have the points, A (2, 3, 4),B (– 1, – 2, 1),C (5, 8, 7);

And as we can find the direction ratios of the line joining the points $(x_{1},y_{1},z_{1}) \ and\ (x_{2},y_{2},z_{2})$ is given by $x_{2}-x_{1}, y_{2}-y_{1}, \ and\ z_{2}-z_{1}.$

The direction ratios of AB are $(-1-2), (-2-3),\ and\ (1-4)$ i.e., $-3,\ -5,\ and\ -3$

The direction ratios of BC are $(5-(-1)), (8-(-2)),\ and\ (7-1)$ i.e., $6,\ 10,\ and\ 6$ .

We can see that the direction ratios of AB and BC are proportional to each other and is -2 times.

$\therefore$ AB is parallel to BC. and as point B is common to both AB and BC,

Hence the points A, B and C are collinear.

Question:5 Find the direction cosines of the sides of the triangle whose vertices are (3, 5, – 4), (– 1, 1, 2) and (– 5, – 5, – 2).

Given vertices of the triangle $\triangle ABC$ (3, 5, – 4), (– 1, 1, 2) and (– 5, – 5, – 2).

Finding each side direction ratios;

$\Rightarrow$ Direction ratios of side AB are $(-1-3), (1-5),\ and\ (2-(-4))$ i.e.,

$-4,-4,\ and\ 6.$

Therefore its direction cosines values are;

$\frac{-4}{\sqrt{(-4)^2+(-4)^2+(6)^2}},\ \frac{-4}{\sqrt{(-4)^2+(-4)^2+(6)^2}},\ \frac{6}{\sqrt{(-4)^2+(-4)^2+(6)^2}}$ $or\ \frac{-4}{2\sqrt{17}},\frac{-4}{2\sqrt{17}},\frac{6}{2\sqrt{17}}\ or\ \frac{-2}{\sqrt{17}},\frac{-2}{\sqrt{17}},\frac{3}{\sqrt{17}}$

SImilarly for side BC;

$\Rightarrow$ Direction ratios of side BC are $(-5-(-1)), (-5-1),\ and\ (-2-2)$ i.e.,

$-4,-6,\ and\ -4.$

Therefore its direction cosines values are;

$\frac{-4}{\sqrt{(-4)^2+(-6)^2+(-4)^2}},\ \frac{-6}{\sqrt{(-4)^2+(-6)^2+(-4)^2}},\ \frac{-4}{\sqrt{(-4)^2+(-6)^2+(-4)^2}}$ $or\ \frac{-4}{2\sqrt{17}},\frac{-6}{2\sqrt{17}},\frac{-4}{2\sqrt{17}}\ or\ \frac{-2}{\sqrt{17}},\frac{-3}{\sqrt{17}},\frac{-2}{\sqrt{17}}$

$\Rightarrow$ Direction ratios of side CA are $(-5-3), (-5-5),\ and\ (-2-(-4))$ i.e.,

$-8,-10,\ and\ 2.$

Therefore its direction cosines values are;

$\frac{-8}{\sqrt{(-8)^2+(10)^2+(2)^2}},\ \frac{-5}{\sqrt{(-8)^2+(10)^2+(2)^2}},\ \frac{2}{\sqrt{(-8)^2+(10)^2+(2)^2}}$ $or\ \frac{-8}{2\sqrt{42}},\frac{-10}{2\sqrt{42}},\frac{2}{2\sqrt{42}}\ or\ \frac{-4}{\sqrt{42}},\frac{-5}{\sqrt{42}},\frac{1}{\sqrt{42}}$

## More About NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1

The NCERT book exercise 11.1 Class 12 Maths have 5 questions. The problems in the NCERT solutions for Class 12 Maths chapter 11 exercise 11.1 are to find the direction cosines and one among the five questions in the Class 12th Maths chapter 11 exercise 11.1 is to find whether the given points are collinear or not. No multiple-choice questions are given in the Class 12 Maths chapter 11 exercise 11.1.

Also Read| Three Dimensional Geometry Class 12th Notes

## Benefits of NCERT Solutions for Class 12 Maths Chapter 11 Exercise 11.1

• The exercise 11.1 Class 12 Maths clears the concepts of direction cosines and ratios.

• All the questions of NCERT syllabus for Class 12 Maths chapter 11 exercise 11.1 are important and will be useful in CBSE board exam preparation.

Also see-

• NCERT Exemplar Solutions Class 12 Maths Chapter 11

• NCERT Solutions for Class 12 Maths Chapter 11

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