Class 12 Maths Chapter 11 Three Dimensional Geometry – Exercise 11.3 NCERT Solutions

Introduction

Exercise 11.3 focuses on the equations of a line in space. Students learn how to represent a line in vector form and Cartesian form, and how to solve problems involving points, direction ratios, and direction cosines. This exercise is essential for understanding the geometry of lines in three‑dimensional space.

Key Concepts

1. Vector Form of Line: Line through point $A(x_1,y_1,z_1)$ and parallel to vector $\vec{b}=(a,b,c)$:

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

where $\vec{a}=x_1\widehat{i}+y_1\widehat{j}+z_1\widehat{k}$.

2. Cartesian Form of Line:

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

3. Direction Ratios (DRs): $a,b,c$ are direction ratios of the line.

4. Relation Between Vector and Cartesian Form: Both represent the same line in different notations.

Students Frequently Make Mistakes

• Forgetting to substitute point coordinates correctly.

• Mixing up vector and Cartesian forms.

• Errors in identifying direction ratios.

• Skipping parameter $\lambda$ in vector form.

• Misinterpreting geometric meaning of line equations.

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

Q1. Find vector equation of line through $A(1,2,3)$ and parallel to vector $\vec{b}=2\widehat{i}+3\widehat{j}+\widehat{k}$.

$$\vec{r}=(\widehat{i}+2\widehat{j}+3\widehat{k})+\lambda (2\widehat{i}+3\widehat{j}+\widehat{k})$$

Q2. Find Cartesian equation of line through $A(1,2,3)$ and parallel to vector $(2,3,1)$.

$$\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{1}$$

Q3. Find vector equation of line through $P(2,-1,3)$ and parallel to vector $(1,2,-2)$.

$$\vec{r}=(2\widehat{i}-\widehat{j}+3\widehat{k})+\lambda (\widehat{i}+2\widehat{j}-2\widehat{k})$$

Q4. Find Cartesian equation of line through $P(2,-1,3)$ and parallel to vector $(1,2,-2)$.

$$\frac{x-2}{1}=\frac{y+1}{2}=\frac{z-3}{-2}$$

Q5. Find vector equation of line through $A(0,0,0)$ and parallel to vector $(1,1,1)$.

$$\vec{r}=\lambda (\widehat{i}+\widehat{j}+\widehat{k})$$

Q6. Find Cartesian equation of line through $A(0,0,0)$ and parallel to vector $(1,1,1)$.

$$\frac{x}{1}=\frac{y}{1}=\frac{z}{1}$$

Q7. Find vector equation of line through $A(3,4,5)$ and parallel to vector $(2,-1,3)$.

$$\vec{r}=(3\widehat{i}+4\widehat{j}+5\widehat{k})+\lambda (2\widehat{i}-\widehat{j}+3\widehat{k})$$

Q8. Find Cartesian equation of line through $A(3,4,5)$ and parallel to vector $(2,-1,3)$.

$$\frac{x-3}{2}=\frac{y-4}{-1}=\frac{z-5}{3}$$

Q9. Find vector equation of line through $P(1,2,3)$ and parallel to vector $(0,1,-1)$.

$$\vec{r}=(\widehat{i}+2\widehat{j}+3\widehat{k})+\lambda (0\widehat{i}+\widehat{j}-\widehat{k})$$

Q10. Find Cartesian equation of line through $P(1,2,3)$ and parallel to vector $(0,1,-1)$.

$$\frac{x-1}{0}=\frac{y-2}{1}=\frac{z-3}{-1}$$

(Simplified: $x=1,\frac{y-2}{1}=\frac{z-3}{-1}$)

FAQs (10)

FAQ1. What is vector equation of line? $\vec{r}=\vec{a}+\lambda \vec{b}$.

FAQ2. What is Cartesian equation of line? $\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}$.

FAQ3. What are direction ratios? Numbers proportional to components of line vector.

FAQ4. How to convert vector form to Cartesian form? Compare coefficients of $\widehat{i},\widehat{j},\widehat{k}$.

FAQ5. What is vector equation of line through origin and parallel to $(1,1,1)$? $\vec{r}=\lambda (\widehat{i}+\widehat{j}+\widehat{k})$.

FAQ6. What is Cartesian equation of line through origin and parallel to $(1,1,1)$? $\frac{x}{1}=\frac{y}{1}=\frac{z}{1}$.

FAQ7. What is vector equation of line through $(2,-1,3)$ and parallel to $(1,2,-2)$? $\vec{r}=(2\widehat{i}-\widehat{j}+3\widehat{k})+\lambda (\widehat{i}+2\widehat{j}-2\widehat{k})$.

FAQ8. What is Cartesian equation of line through $(2,-1,3)$ and parallel to $(1,2,-2)$? $\frac{x-2}{1}=\frac{y+1}{2}=\frac{z-3}{-2}$.

FAQ9. What is condition for line parallel to vector? Direction ratios equal to vector components.

FAQ10. Why is Exercise 11.3 important? It builds foundation for equations of lines in 3D geometry.

Conclusion

Exercise 11.3 has 10 solved questions and 10 FAQs that strengthen your understanding of vector and Cartesian equations of lines in 3D geometry. This builds the foundation for advanced study of lines and planes in Class 12 Maths.

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