NCERT Solutions for Class 12 Mathematics Chapter 11 Three Dimensional Geometry

This worksheet provides complete and accurate NCERT Solutions for Class 12 Mathematics Chapter 11 Three Dimensional Geometry. This chapter introduces students to important concepts of three dimensional space using vector algebra, making geometry more clear and systematic. Students learn how to find direction cosines, direction ratios, equations of lines, angles between lines, and shortest distances between lines. This chapter is important as it builds strong foundations for higher mathematics and helps in solving real-world spatial problems. These solutions strictly follow NCERT guidelines and help students understand each concept step by step.

Chapter summary: stories, poems & themes

This chapter is concept-based and focuses on mathematical understanding rather than stories or poems. It explains how lines behave in three dimensional space using vectors. Students learn to visualize space, understand directions, and solve problems related to lines and distances. The chapter is analytical and focuses on applying formulas and concepts correctly.

What this NCERT chapter covers?

• Understanding direction cosines and direction ratios of a line 
• Finding direction cosines using two given points 
• Writing vector and Cartesian equations of a line 
• Calculating angle between two lines 
• Understanding parallel, perpendicular, and skew lines 
• Finding shortest distance between two lines 
• Applying vector algebra in three dimensional geometry 

How to use these NCERT solutions?

Students should first attempt all questions from the worksheet on their own. After solving, they can refer to these NCERT solutions to check accuracy and correct mistakes. Parents and teachers can guide students by explaining steps where needed. These solutions follow the exact NCERT sequence, making revision easy and structured. Regular practice using these solutions improves understanding and helps in exams.

Student tips & learning tricks

• Always remember that direction cosines satisfy l² + m² + n² = 1 
• Do not confuse direction ratios with direction cosines 
• Carefully apply formulas for angle and shortest distance 
• Check calculations while solving vector equations 
• Practice diagrams to understand spatial concepts clearly 

Why NCERT solutions are important?

NCERT solutions help students build strong conceptual clarity as they follow the official CBSE syllabus. They ensure correct methods and accurate answers, which are essential for exams. These solutions improve confidence, reduce mistakes, and help students perform better in assessments. They also prepare students for higher-level mathematics studies.

Complete answer key – NCERT solutions

Exercise 11.1

1. 
l = 0 
m = -1/√2 
n = 1/√2 

2. 
l = 1/√3 
m = 1/√3 
n = 1/√3 

3. 
l = -9/11 
m = 6/11 
n = -2/11 

4. 
Points are collinear 

5. 
AB: (-2/√17, -2/√17, 3/√17) 
BC: (-2/√17, -3/√17, -2/√17) 
CA: (4/√42, 5/√42, -1/√42) 

Exercise 11.2

1. 
Lines are mutually perpendicular 

2. 
Lines are perpendicular 

3. 
Lines are parallel 

4. 
Vector form: r = (i + 2j + 3k) + λ(3i + 2j - 2k) 
Cartesian form: (x - 1)/3 = (y - 2)/2 = (z - 3)/(-2) 

5. 
Vector form: r = (2i - 4j + k) + λ(2i + j - k) 
Cartesian form: (x - 2)/2 = (y + 4)/1 = (z - 1)/(-1) 

6. 
(x + 2)/3 = (y - 4)/5 = (z + 5)/6 

7. 
r = (5i - 4j + 6k) + λ(3i + 7j + 2k) 

8. 
(i) cosθ = 16/21 
(ii) cosθ = 6/√150 

9. 
(i) cosθ = 4/(7√10) 
(ii) cosθ = 4/√798 

10. 
p = -11/5 

11. 
Lines are perpendicular 

12. 
|(b₁ × b₂) · (a₂ - a₁)| / |b₁ × b₂| 

13. 
2√3 

14. 
√14 

15. 
√6 

Miscellaneous Exercise

1. 
Angle = 90° 

2. 
r = λi 

3. 
k = 1 

4. 
√21 

5. 
r = (i + 2j - 4k) + λ(2i + j + k) 

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