Example 12
Last updated at March 11, 2017 by Teachoo
Transcript
Example 12 Find the distance between the lines 𝑙1 and 𝑙2 given by 𝑟 = 𝑖 + 2 𝑗 – 4 𝑘 + 𝜆(2 𝒊 + 3 𝒋 + 6 𝒌 ) and 𝑟 = 3 𝑖 + 3 𝑗 − 5 𝑘 + μ (2 𝒊 + 3 𝒋 + 6 𝒌) Distance between two parallel lines with vector equations 𝑟 = 𝑎1 + 𝜆 𝑏 and 𝑟 = 𝑎2 + μ 𝑏 is 𝑏 × ( 𝑎2 − 𝑎1) 𝑏 Now, ( 𝒂𝟐 − 𝒂𝟏) = (3 𝑖 + 3 𝑗 − 5 𝑘) − (1 𝑖 + 2 𝑗 − 4 𝑘) = (3 − 1) 𝑖 + (3 − 2) 𝑗 + ( − 5 + 4) 𝑘 = 2 𝒊 + 1 𝒋 − 1 𝒌 Magnitude of 𝑏 = 22 + 32 + 62 𝒃 = 4+9+36 = 49 = 7 Also, 𝒃 × ( 𝒂𝟐 − 𝒂𝟏) = 𝑖 𝑗 𝑘23621−1 = 𝑖 3×−1−(1×6) − 𝑗 2×−1−(2×6) + 𝑘 2×1−(2×3) = 𝑖 −3−6 − 𝑗 −2−12 + 𝑘 2−6 = 𝑖 (–9) − 𝑗 (–14) + 𝑘(−4) = −𝟗 𝒊 + 14 𝒋 − 4 𝒌 Now, 𝒃 × ( 𝒂𝟐 − 𝒂𝟏) = −92+ 142+ −42 = 81+196+16 = 𝟐𝟗𝟑 So, distance = 𝑏 × ( 𝑎2 − 𝑎1) 𝑏 = 2937 = 𝟐𝟗𝟑𝟕 Therefore, the distance between the given two parallel lines is 2937.
Examples
Example 1
Example, 2
Example, 3 Important
Example, 4
Example, 5
Example, 6 Important
Example, 7
Example 8
Example, 9 Important
Example 10
Example 11
Example 12 Important You are here
Example 13
Example 14
Example 15
Example 16
Example 17
Example 18
Example 19
Example 20 Important
Example 21 Important
Example 22
Example 23 Important
Example 24 Important
Example, 25 Important
Example 26
Example 27 Important
Example 28
Example 29 Important
Example 30 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 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here https://www.teachoo.com/premium/maths-and-science-classes/.