Class 12 Maths Chapter 11 Three Dimensional Geometry – Exercise 11.2 NCERT Solutions

Introduction

Exercise 11.2 focuses on the direction cosines and direction ratios of a line in 3D geometry. Students learn how to calculate angles made by a line with coordinate axes, find direction cosines, and verify relationships between them. This exercise is essential for understanding equations of lines and planes in three‑dimensional space.

Key Concepts

1. Direction Ratios (DRs): If line has vector $\vec{a}=(x,y,z)$, then $x,y,z$ are direction ratios.

2. Direction Cosines (DCs): If line makes angles $\alpha ,\beta ,\gamma$ with x‑, y‑, z‑axes, then

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

3. Relation Between DCs:

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

4. Conversion: If DRs are $a,b,c$, then

$$l=\frac{a}{\sqrt{a^2+b^2+c^2}},m=\frac{b}{\sqrt{a^2+b^2+c^2}},n=\frac{c}{\sqrt{a^2+b^2+c^2}}$$

Students Frequently Make Mistakes

• Forgetting to normalize direction ratios to get cosines.

• Errors in squaring and summing cosines.

• Misinterpreting angles with axes.

• Skipping verification of $l^2+m^2+n^2=1$.

• Confusing DRs with DCs.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. Find direction cosines of line with DRs $1,2,2$.

$$l=\frac{1}{\sqrt{1+4+4}}=\frac{1}{3},m=\frac{2}{3},n=\frac{2}{3}$$

Q2. Find direction cosines of line with DRs $2,-1,2$.

$$l=\frac{2}{\sqrt{4+1+4}}=\frac{2}{3},m=\frac{-1}{3},n=\frac{2}{3}$$

Q3. Find direction cosines of line with DRs $3,4,12$.

$$l=\frac{3}{13},m=\frac{4}{13},n=\frac{12}{13}$$

Q4. Find direction cosines of line with DRs $1,1,1$.

$$l=m=n=\frac{1}{\sqrt{3}}$$

Q5. Find direction cosines of line with DRs $2,2,1$.

$$l=\frac{2}{3},m=\frac{2}{3},n=\frac{1}{3}$$

Q6. Find direction cosines of line with DRs $1,-2,2$.

$$l=\frac{1}{3},m=\frac{-2}{3},n=\frac{2}{3}$$

Q7. Find direction cosines of line with DRs $0,1,\sqrt{3}$.

$$l=0,m=\frac{1}{2},n=\frac{\sqrt{3}}{2}$$

Q8. Find direction cosines of line with DRs $1,0,1$.

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

Q9. Find direction cosines of line with DRs $2,3,6$.

$$l=\frac{2}{7},m=\frac{3}{7},n=\frac{6}{7}$$

Q10. Verify relation $l^2+m^2+n^2=1$ for DRs $1,2,2$.

$$l=\frac{1}{3},m=\frac{2}{3},n=\frac{2}{3}$$

$$l^2+m^2+n^2=\frac{1}{9}+\frac{4}{9}+\frac{4}{9}=1$$

FAQs (10)

FAQ1. What are direction ratios? Numbers proportional to components of line vector.

FAQ2. What are direction cosines? Cosines of angles made with coordinate axes.

FAQ3. What is relation between DCs? $l^2+m^2+n^2=1$.

FAQ4. How to convert DRs to DCs? Divide each DR by $\sqrt{a^2+b^2+c^2}$.

FAQ5. What are DCs of line with DRs $1,1,1$? $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$.

FAQ6. What are DCs of line with DRs $2,-1,2$? $\frac{2}{3},-\frac{1}{3},\frac{2}{3}$.

FAQ7. What are DCs of line with DRs $3,4,12$? $\frac{3}{13},\frac{4}{13},\frac{12}{13}$.

FAQ8. What is geometric meaning of DCs? They represent orientation of line in 3D space.

FAQ9. When are vectors coplanar? If scalar triple product = 0 (from previous exercise).

FAQ10. Why is Exercise 11.2 important? It builds foundation for equations of lines and planes in 3D.

Conclusion

Exercise 11.2 has 10 solved questions and 10 FAQs that strengthen your understanding of direction cosines and direction ratios in 3D geometry. This builds the foundation for equations of lines and planes in Class 12 Maths.

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