Last updated at May 29, 2018 by Teachoo

Transcript

Misc 9 Find the shortest distance between lines 𝑟 = 6 𝑖 + 2 𝑗 + 2 𝑘 + 𝜆 ( 𝑖 – 2 𝑗 + 2 𝑘) and 𝑟 = –4 𝑖 – 𝑘 + 𝜇 (3 𝑖 – 2 𝑗 – 2 𝑘) . Shortest distance between lines with vector equations 𝑟 = 𝑎1 + 𝜆 𝑏1 and 𝑟 = 𝑎2 + 𝜇 𝑏2 is 𝒃𝟏 × 𝒃𝟐 . 𝒂𝟐 − 𝒂𝟏 𝒃𝟏 × 𝒃𝟐 Now, ( 𝒂𝟐 − 𝒂𝟏) = (−4 𝑖 + 0 𝑗 − 1 𝑘) − (6 𝑖 + 2 𝑗 + 2 𝑘) = (−4 − 6) 𝑖 + (0 − 2) 𝑗 + (−1 − 2) 𝑘 = − 10 𝒊 − 2 𝒋 − 3 𝒌 ( 𝒃𝟏 × 𝒃𝟐) = 𝑖 𝑗 𝑘1 −223−2−2 = 𝑖 −2×−2−(−2×2) − 𝑗 1×−2−(3×2) + 𝑘 1×−2−(3×−2) = 𝑖 4+4 − 𝑗 −2−6 + 𝑘 −2+6 = 𝑖 (8) − 𝑗 (−8) + 𝑘(4) = 8 𝒊 + 8 𝒋 + 4 𝒌 Magnitude of 𝑏1 × 𝑏2 = 82+ 82+ 42 𝒃𝟏 × 𝒃𝟐 = 64+64+16 = 144 = 𝟏𝟐 Also, 𝒃𝟏× 𝒃𝟐 . 𝒂𝟐 − 𝒂𝟏 = (8 𝑖 + 8 𝑗 + 4 𝑘).(− 10 𝑖 − 2 𝑗 − 3 𝑘) = (8 × − 10) + (8 × − 2) + (4 × − 3) = − 80 + (−16) + (-12) = − 108 Shortest distance = 𝑏1 × 𝑏2 . 𝑎2 − 𝑎1 𝑏1 × 𝑏2 = −10812 = −9 = 9 Therefore, the shortest distance between the given two lines is 9.

Chapter 11 Class 12 Three Dimensional Geometry

Example, 3
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Ex 11.1, 2 Important

Example, 6 Important

Example, 9 Important

Example 12 Important

Ex 11.2, 5 Important

Ex 11.2, 11 Important

Ex 11.2, 12 Important

Ex 11.2, 14 Important

Ex 11.2, 15 Important

Ex 11.2, 17 Important

Example 20 Important

Example 21 Important

Example 23 Important

Example 24 Important

Example, 25 Important

Ex 11.3, 4 Important

Ex 11.3, 11 Important

Ex 11.3, 12 Important

Ex 11.3, 14 Important

Example 27 Important

Example 29 Important

Example 30 Important

Misc 6 Important

Misc 9 Important You are here

Misc 14 Important

Misc 18 Important

Misc 20 Important

Misc 21 Important

Class 12

Important Question for exams Class 12

- Chapter 1 Class 12 Relation and Functions
- Chapter 2 Class 12 Inverse Trigonometric Functions
- Chapter 3 Class 12 Matrices
- Chapter 4 Class 12 Determinants
- Chapter 5 Class 12 Continuity and Differentiability
- Chapter 6 Class 12 Application of Derivatives
- Chapter 7 Class 12 Integrals
- Chapter 8 Class 12 Application of Integrals
- Chapter 9 Class 12 Differential Equations
- Chapter 10 Class 12 Vector Algebra
- Chapter 11 Class 12 Three Dimensional Geometry
- Chapter 12 Class 12 Linear Programming
- Chapter 13 Class 12 Probability

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.