Class 11 Maths Chapter 11 Three Dimensional Geometry – Exercise 11.2 NCERT Solutions

Introduction

Exercise 11.2 focuses on direction cosines and direction ratios of a line in 3D geometry. You will learn how to represent the orientation of a line using cosines of angles with coordinate axes, calculate direction ratios, and apply these concepts to problems involving angles between lines.

Key Concepts

1. Direction Cosines (l, m, n): Cosines of angles made by a line with x‑axis, y‑axis, and z‑axis.

$$l=\cos \alpha ,m=\cos \beta ,n=\cos \gamma$$

2. Property of Direction Cosines:

$$l^2+m^2+n^2=1$$

3. Direction Ratios (a, b, c): Any set of numbers proportional to direction cosines.

4. Angle Between Two Lines: If direction cosines are $(l_1,m_1,n_1)$ and $(l_2,m_2,n_2)$:

$$\cos \theta =l_1{l}_2+m_1{m}_2+n_1{n}_2$$

NCERT Questions with Solutions (10)

Q1. Find direction cosines of line equally inclined to axes.

$$l=m=n\text{and}{l}^2+m^2+n^2=1$$

$$3l^2=1⟹l=m=n=\frac{1}{\sqrt{3}}$$

Q2. Find direction cosines of line making angles ${90}^{\circ },{45}^{\circ },{60}^{\circ }$ with axes.

$$l=\cos {90}^{\circ }=0,m=\cos {45}^{\circ }=\frac{1}{\sqrt{2}},n=\cos {60}^{\circ }=\frac{1}{2}$$

Q3. Find direction cosines of line making angles ${60}^{\circ },{60}^{\circ },{60}^{\circ }$.

$$l=m=n=\cos {60}^{\circ }=\frac{1}{2}$$

Check: $l^2+m^2+n^2=\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=\frac{3}{4}\mathrm{\ne }1$. Hence such a line does not exist.

Q4. Find direction cosines of line making angles $0^{\circ },{90}^{\circ },{90}^{\circ }$.

$$l=\cos 0^{\circ }=1,m=\cos {90}^{\circ }=0,n=\cos {90}^{\circ }=0$$

Q5. Find direction cosines of line making angles ${45}^{\circ },{45}^{\circ },{90}^{\circ }$.

$$l=\frac{1}{\sqrt{2}},m=\frac{1}{\sqrt{2}},n=0$$

Q6. Find angle between lines with direction cosines (1,0,0) and (0,1,0).

$$\cos \theta =1\cdot 0+0\cdot 1+0\cdot 0=0⟹\theta ={90}^{\circ }$$

Q7. Find angle between lines with direction cosines (1,0,0) and $(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$.

$$\cos \theta =1\cdot \frac{1}{\sqrt{2}}+0+0=\frac{1}{\sqrt{2}}$$

$\theta ={45}^{\circ }$

Q8. Find angle between lines with direction cosines (0,1,0) and (0,0,1).

$$\cos \theta =0+0+0=0⟹\theta ={90}^{\circ }$$

Q9. Find angle between lines with direction cosines (1,0,0) and (1,1,1)/$\sqrt{3}$.

$$\cos \theta =\frac{1}{\sqrt{3}}⟹\theta ={\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$$

Q10. Find angle between lines with direction cosines (0,1,0) and (1,1,0)/$\sqrt{2}$.

$$\cos \theta =\frac{1}{\sqrt{2}}⟹\theta ={45}^{\circ }$$

FAQs (10)

1. What are direction cosines? Cosines of angles made by line with coordinate axes.

2. What is property of direction cosines? $l^2+m^2+n^2=1$.

3. What are direction ratios? Numbers proportional to direction cosines.

4. How to find angle between two lines? Use formula $\cos \theta =l_1{l}_2+m_1{m}_2+n_1{n}_2$.

5. Can direction cosines be negative? Yes, depending on orientation.

6. What is relation between direction cosines and ratios? Ratios are proportional to cosines.

7. What is angle between x‑axis and y‑axis? ${90}^{\circ }$.

8. What is angle between x‑axis and line equally inclined to axes? ${\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$.

9. What is angle between y‑axis and z‑axis? ${90}^{\circ }$.

10. Why is this exercise important? It builds foundation for vectors and line equations in 3D geometry.

Conclusion

Exercise 11.2 has 10 solved questions and 10 FAQs that strengthen your understanding of direction cosines, direction ratios, and angles between lines in 3D geometry. This builds the foundation for vectors and advanced 3D geometry in Class 11 Maths.

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New Syllabus-Class 11 Maths Chapter 11 Three Dimensional Geometry – Exercise 11.2 NCERT