Class 12 Maths Chapter 11 Three Dimensional Geometry – Exercise 11.5 NCERT Solutions

Introduction

Exercise 11.5 focuses on the angle between two lines, angle between a line and a plane, and angle between two planes in three‑dimensional geometry. Students learn how to apply dot product and normal vector concepts to calculate these angles. This exercise is crucial for solving advanced problems in 3D geometry.

Key Concepts

1. Angle Between Two Lines: If direction ratios of lines are $(a_1,b_1,c_1)$ and $(a_2,b_2,c_2)$, then

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

2. Angle Between Line and Plane: If line has direction ratios $(a,b,c)$ and plane has normal vector $(l,m,n)$, then

$$\sin \theta =\frac{\mathrm{\mid }al+bm+cn\mathrm{\mid }}{\sqrt{a^2+b^2+c^2}\cdot \sqrt{l^2+m^2+n^2}}$$

3. Angle Between Two Planes: If normals are $(a_1,b_1,c_1)$ and $(a_2,b_2,c_2)$, then

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

Students Frequently Make Mistakes

• Confusing formulas for line–line and plane–plane angles.

• Forgetting to normalize direction ratios.

• Errors in applying sine vs cosine formula.

• Misinterpreting perpendicularity condition.

• Skipping absolute value in numerator.

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

Q1. Find angle between lines with DRs $(1,2,2)$ and $(2,1,-2)$.

$$\cos \theta =\frac{1\cdot 2+2\cdot 1+2\cdot (-2)}{\sqrt{1+4+4}\cdot \sqrt{4+1+4}}=\frac{2+2-4}{3\cdot 3}=0$$

$\theta ={90}^{\circ }$.

Q2. Find angle between lines with DRs $(1,1,1)$ and $(2,2,2)$.

$$\cos \theta =\frac{1\cdot 2+1\cdot 2+1\cdot 2}{\sqrt{3}\cdot \sqrt{12}}=\frac{6}{\sqrt{36}}=1$$

$\theta =0^{\circ }$.

Q3. Find angle between planes $x+y+z=1$ and $2x+2y+2z=5$. Normals: $(1,1,1)$, $(2,2,2)$. Clearly parallel ⇒ $\theta =0^{\circ }$.

Q4. Find angle between planes $x+2y+3z=4$ and $2x+3y+z=5$. Normals: $(1,2,3)$, $(2,3,1)$.

$$\cos \theta =\frac{1\cdot 2+2\cdot 3+3\cdot 1}{\sqrt{14}\cdot \sqrt{14}}=\frac{11}{14}$$

Q5. Find angle between line with DRs $(1,2,2)$ and plane $x+y+z=0$. Normal of plane: $(1,1,1)$.

$$\sin \theta =\frac{\mathrm{\mid }1\cdot 1+2\cdot 1+2\cdot 1\mathrm{\mid }}{\sqrt{9}\cdot \sqrt{3}}=\frac{5}{3\sqrt{3}}$$

Q6. Find angle between line with DRs $(2,-1,2)$ and plane $x-2y+z=0$. Normal: $(1,-2,1)$.

$$\sin \theta =\frac{\mathrm{\mid }2\cdot 1+(-1)(-2)+2\cdot 1\mathrm{\mid }}{\sqrt{9}\cdot \sqrt{6}}=\frac{6}{3\sqrt{6}}=\frac{2}{\sqrt{6}}$$

Q7. Find angle between planes $x+2y+3z=0$ and $3x+6y+9z=5$. Normals proportional ⇒ planes parallel ⇒ $\theta =0^{\circ }$.

Q8. Find angle between planes $x-y+z=0$ and $y-z=0$. Normals: $(1,-1,1)$, $(0,1,-1)$.

$$\cos \theta =\frac{1\cdot 0+(-1)\cdot 1+1\cdot (-1)}{\sqrt{3}\cdot \sqrt{2}}=\frac{-2}{\sqrt{6}}$$

Q9. Find angle between line with DRs $(1,0,1)$ and plane $x+z=0$. Normal: $(1,0,1)$.

$$\sin \theta =\frac{\mathrm{\mid }1\cdot 1+0\cdot 0+1\cdot 1\mathrm{\mid }}{\sqrt{2}\cdot \sqrt{2}}=\frac{2}{2}=1$$

$\theta ={90}^{\circ }$.

Q10. Find angle between planes $2x-y+z=0$ and $x+y+z=0$. Normals: $(2,-1,1)$, $(1,1,1)$.

$$\cos \theta =\frac{2\cdot 1+(-1)\cdot 1+1\cdot 1}{\sqrt{6}\cdot \sqrt{3}}=\frac{2}{\sqrt{18}}=\frac{2}{3\sqrt{2}}$$

FAQs (10)

FAQ1. What is formula for angle between two lines? $\cos \theta =\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\cdot \sqrt{a_2^2+b_2^2+c_2^2}}$.

FAQ2. What is formula for angle between line and plane? $\sin \theta =\frac{\mathrm{\mid }al+bm+cn\mathrm{\mid }}{\sqrt{a^2+b^2+c^2}\cdot \sqrt{l^2+m^2+n^2}}$.

FAQ3. What is formula for angle between two planes? Same as angle between normals.

FAQ4. When are two lines perpendicular? If dot product of DRs = 0.

FAQ5. When are two planes parallel? If normals are proportional.

FAQ6. What is angle between line $(1,1,1)$ and plane $x+y+z=0$? 90°.

FAQ7. What is angle between planes $x+2y+3z=0$ and $2x+3y+z=0$? $\cos \theta =\frac{11}{14}$.

FAQ8. What is angle between line $(1,0,1)$ and plane $x+z=0$? 90°.

FAQ9. What is angle between planes with normals $(1,-1,1)$ and $(0,1,-1)$? $\cos \theta =\frac{-2}{\sqrt{6}}$.

FAQ10. Why is Exercise 11.5 important? It builds mastery of angles between lines and planes in 3D geometry.

Conclusion

Exercise12.5 has 10 solved questions and 10 FAQs that strengthen your understanding of inverse trigonometric functions, principal values, and identities. This builds the foundation for calculus in Class 12 Maths.

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