Get live Maths 1-on-1 Classs - Class 6 to 12
Miscellaneous
Misc 2
Misc 3 You are here
Misc 4 Important
Misc 5 Important
Misc 6 Important
Misc 7 Deleted for CBSE Board 2023 Exams
Misc 8 Important Deleted for CBSE Board 2023 Exams
Misc 9 Important
Misc 10 Deleted for CBSE Board 2023 Exams
Misc 11 Important Deleted for CBSE Board 2023 Exams
Misc 12 Important Deleted for CBSE Board 2023 Exams
Misc 13 Important Deleted for CBSE Board 2023 Exams
Misc 14 Important Deleted for CBSE Board 2023 Exams
Misc 15 Important Deleted for CBSE Board 2023 Exams
Misc 16 Deleted for CBSE Board 2023 Exams
Misc 17 Important Deleted for CBSE Board 2023 Exams
Misc 18 Important Deleted for CBSE Board 2023 Exams
Misc 19 Deleted for CBSE Board 2023 Exams
Misc 20 Important
Misc 21 Important Deleted for CBSE Board 2023 Exams
Misc 22 (MCQ) Important Deleted for CBSE Board 2023 Exams
Misc 23 (MCQ) Important Deleted for CBSE Board 2023 Exams
Miscellaneous
Last updated at March 22, 2023 by Teachoo
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°