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.

Miscellaneous

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Misc 2

Misc 3

Misc 4 Important

Misc 5 Important

Misc 6 Important

Misc 7

Misc 8 Important

Misc 9 Important You are here

Misc 10

Misc 11 Important

Misc 12 Important

Misc 13 Important

Misc 14 Important

Misc 15 Important

Misc 16 Important

Misc 17 Important

Misc 18 Important

Misc 19 Important

Misc 20 Important

Misc 21 Important

Misc 22 Important

Misc 23 Important

Chapter 11 Class 12 Three Dimensional Geometry

Serial order wise

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.