Last updated at Dec. 8, 2016 by Teachoo

Transcript

Misc 3 Find the angle between the lines whose direction ratios are a, b, c and b − c, c − a, a − b. Angle between the lines with direction ratios a1, b1, c1 and a2, b2, c2 is given by cos θ = 𝒂𝟏 𝒂𝟐 + 𝒃𝟏 𝒃𝟐 + 𝒄𝟏 𝒄𝟐 𝒂𝟏𝟐 + 𝒃𝟏𝟐 + 𝒄𝟏𝟐 𝒂𝟏𝟐 + 𝒃𝟏𝟐 + 𝒄𝟏𝟐 Given, 𝑎1 = 𝑎, 𝑏1 = 𝑏, c1 = c and 𝑎2 = 𝑏 − 𝑐, 𝑏2 = 𝑐 − 𝑎, c2 = a – b So, cos θ = 𝑎 𝑏 − 𝑐 + 𝑏 𝑐 − 𝑎 + 𝑐(𝑎 − 𝑏) 𝑎2 + 𝑏2 + 𝑐2 𝑏 − 𝑐2 + 𝐶 − 𝑎2 + 𝑎 − 𝑏2 = 𝑎𝑏 − 𝑎𝑐 + 𝑏𝑐 − 𝑎𝑏 + 𝑐𝑎 − 𝑏𝑐 𝑎2 + 𝑏2 + 𝑐2 𝑏2 + 𝑐2 − 2𝑏𝑐 + 𝑐2 + 𝑎2 − 2𝑐𝑎 + 𝑎2 + 𝑏2 − 2𝑎𝑏 = 0 ∴ cos θ = 0 So, θ = 90° Therefore, angle between the given pair of lines is 90°

Miscellaneous

Misc 1
Important

Misc 2

Misc 3 Deleted for CBSE Board 2021 Exams only You are here

Misc 4 Important

Misc 5 Important Deleted for CBSE Board 2021 Exams only

Misc 6 Important

Misc 7

Misc 8 Important

Misc 9 Important

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 10 years. He provides courses for Maths and Science at Teachoo.