



Get live Maths 1-on-1 Classs - Class 6 to 12
Miscellaneous
Misc 2
Misc 3
Misc 4 Important
Misc 5 Important You are here
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 5 If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (– 4, 3, – 6) and (2, 9, 2) respectively, then find the angle between the lines AB and CD.Angle between a pair of lines having direction ratios 𝑎1, 𝑏1, c1 and 𝑎_2 , 𝑏_2, 𝑐_2 is given by cos θ = |(𝒂_𝟏 𝒂_𝟐 + 𝒃_𝟏 𝒃_𝟐 + 𝒄𝟏𝒄𝟐)/(√(〖𝒂_𝟏〗^𝟐 + 〖𝒃_𝟏〗^𝟐+〖 𝒄_𝟏〗^𝟐 ) √(〖𝒂_𝟐〗^𝟐 + 〖𝒃_𝟐〗^𝟐+〖 𝒄_𝟐〗^𝟐 ))| A line passing through A (𝑥_1, 𝑦_1, 𝑧_1) and B (𝑥_2, 𝑦_2, 𝑧_2) has direction ratios (𝑥_2 − 𝑥_1), (𝑦_2 − 𝑦_1), (𝑧_2 − 𝑧_1) AB A (1, 2, 3) , B (4, 5, 7) Direction ratios of AB (4 − 1), (5 − 2),(7 − 3) = 3, 3, 4 ∴ 𝒂1 = 3, 𝒃1 = 3, 𝒄1 = 4 CD C (−4, 3, −6) ,D (2, 9, 2) Direction ratios of CD (2 − (–4)), (9 − 3),(2 – (–6)) = 6, 6, 8 ∴ 𝒂2 = 6, 𝒃2 = 6, 𝒄2 = 8 Now, cos θ = |(𝑎_1 𝑎_2 + 𝑏_1 𝑏_2 + 𝑐1𝑐2)/(√(〖𝑎_1〗^2 + 〖𝑏_1〗^2+〖 𝑐_1〗^2 ) √(〖𝑎_2〗^2 + 〖𝑏_2〗^2+〖 𝑐_2〗^2 ))| cos θ = |(3 × 6 + 3 × 6 + 4 × 8 )/(√(32 + 32 + 42) √(62 + 62 + 82))| = |(18 + 18 + 32 )/(√(9 + 9 + 16) √(36 + 36 + 64))| = |68/(√34 √136)| = |68/(√34 √(4 × 34))| = |68/(√34 × √4 × √34)| = |68/(√34 × √34× √4)| = |68/(34 × 2 )| = |68/68| = 1 ∴ cos θ = 1 So, θ = 0° Therefore, angle between AB and CD is 0° .