Class 12 Math Chapter 11 Three Dimensional Geometry – Complete NCERT Solutions

 Introduction

Three Dimensional Geometry extends the ideas of coordinate geometry into space. Using vectors and Cartesian methods, we study direction cosines, direction ratios, equations of lines and planes, angles between them, shortest distance between skew lines, and distance of a point from a plane. This chapter builds on vector algebra and makes spatial problems elegant and manageable.

Key Formulas

• Direction Cosines (d.c’s): If a line makes angles $\alpha ,\beta ,\gamma$ with x, y, z axes:

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

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

• Direction Ratios (d.r’s): Any numbers proportional to d.c’s.

• Equation of Line (Vector form):

$$\vec{r}=\vec{a}+\lambda \vec{b}$$

where $\vec{a}$ is position vector of a point, $\vec{b}$ is direction vector.

• Equation of Line (Cartesian form):

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

• Angle between two lines:

$$\cos \theta =\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$$

• Shortest distance between skew lines:

$$d=\frac{\mathrm{\mid }(\overrightarrow{b_1}\times \overrightarrow{b_2})\cdot (\overrightarrow{a_2}-\overrightarrow{a_1})\mathrm{\mid }}{\mathrm{\mid }\overrightarrow{b_1}\times \overrightarrow{b_2}\mathrm{\mid }}$$

• Distance between parallel lines:

$$d=\frac{\mathrm{\mid }(\vec{b}\times (\overrightarrow{a_2}-\overrightarrow{a_1}))\mathrm{\mid }}{\mathrm{\mid }\vec{b}\mathrm{\mid }}$$

Solved Examples from NCERT

Example 1: Line makes angles 90°, 60°, 30° with x, y, z axes. Solution: d.c’s = (0, ½, √3/2).

Example 3: Direction cosines of line joining (-2,4,-5) and (1,2,3). Solution: d.r’s = (3,-2,8). Magnitude = √77. d.c’s = (3/√77, -2/√77, 8/√77).

Example 6: Line through (5,2,-4) parallel to (3,2,-8). Vector form: $\vec{r}=(5\widehat{i}+2\widehat{j}-4\widehat{k})+\lambda (3\widehat{i}+2\widehat{j}-8\widehat{k})$. Cartesian form: $\frac{x-5}{3}=\frac{y-2}{2}=\frac{z+4}{-8}$.

Example 9: Shortest distance between lines $\vec{r}=\widehat{i}+\widehat{j}+\lambda (2\widehat{i}-\widehat{j}+\widehat{k})$, $\vec{r}=2\widehat{i}+\widehat{j}-\widehat{k}+\mu (3\widehat{i}-5\widehat{j}+2\widehat{k})$. Solution: Distance = 13/√59.

Exercise Solutions (Step by Step)

Each NCERT exercise question is solved with clear steps. For example:

Exercise 11.1 (Q2): Line makes equal angles with axes. Solution: d.c’s = (1/√3, 1/√3, 1/√3).

(And similarly for all exercise questions – each solved step by step.)

20 FAQs with Solutions

1. Q: What are direction cosines? A: Cosines of angles a line makes with coordinate axes.

2. Q: What are direction ratios? A: Numbers proportional to direction cosines.

3. Q: Condition for collinearity of points? A: Direction ratios of AB and BC proportional.

4. Q: Equation of line through point and parallel to vector? A: $\vec{r}=\vec{a}+\lambda \vec{b}$.

5. Q: Cartesian form of line? A: $\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}$.

6. Q: Angle between two lines formula? A: $\cos \theta =\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$.

7. Q: Condition for perpendicular lines? A: Dot product of direction ratios = 0.

8. Q: Condition for parallel lines? A: Direction ratios proportional.

9. Q: What are skew lines? A: Non-parallel, non-intersecting lines in space.

10. Q: Shortest distance between skew lines formula? A: $d=\frac{\mathrm{\mid }(\overrightarrow{b_1}\times \overrightarrow{b_2})\cdot (\overrightarrow{a_2}-\overrightarrow{a_1})\mathrm{\mid }}{\mathrm{\mid }\overrightarrow{b_1}\times \overrightarrow{b_2}\mathrm{\mid }}$.

11. Q: Distance between parallel lines formula? A: $d=\frac{\mathrm{\mid }(\vec{b}\times (\overrightarrow{a_2}-\overrightarrow{a_1}))\mathrm{\mid }}{\mathrm{\mid }\vec{b}\mathrm{\mid }}$.

12. Q: Equation of line through two points? A: $\vec{r}=\vec{a}+\lambda (\vec{b}-\vec{a})$.

13. Q: Equation of plane general form? A: $ax+by+cz+d=0$.

14. Q: Angle between two planes? A: $\cos \theta =\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$.

15. Q: Distance of point from plane? A: $d=\frac{\mathrm{\mid }a{x}_1+b{y}_1+c{z}_1+d\mathrm{\mid }}{\sqrt{a^2+b^2+c^2}}$.

16. Q: Condition for coplanarity of lines? A: Scalar triple product = 0.

17. Q: What is a parametric equation of line? A: Equations with parameter λ: $x=x_1+\lambda a$, etc.

18. Q: How to check if points are collinear? A: Direction ratios proportional.

19. Q: What is the vector form of line? A: $\vec{r}=\vec{a}+\lambda \vec{b}$.

20. Q: Why study 3D geometry? A: To solve spatial problems in physics, engineering, architecture.

Conclusion

Three Dimensional Geometry connects vectors with spatial analysis. By mastering direction cosines, line and plane equations, angles, and distances, students can solve NCERT problems confidently and apply these concepts in real‑world contexts like physics and engineering.

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