Check sibling questions

Ex 11.2, 11 - Class 12 3D Geometry - Find angle between lines

Ex 11.2, 11 - Chapter 11 Class 12 Three Dimensional Geometry - Part 2
Ex 11.2, 11 - Chapter 11 Class 12 Three Dimensional Geometry - Part 3

This video is only available for Teachoo black users

 

 

Maths Crash Course - Live lectures + all videos + Real time Doubt solving!


Transcript

Angle between the pair of lines (𝑥 − 𝑥1)/𝑎1 = (𝑦 − 𝑦1)/𝑏1 = (𝑧 − 𝑧1)/𝑐1 and (𝑥 − 𝑥2)/𝑎2 = (𝑦 − 𝑦2)/𝑏2 = (𝑧 − 𝑧2)/𝑐2 is given by cos θ = |(𝒂_𝟏 𝒂_𝟐 + 𝒃_𝟏 𝒃_𝟐 +〖 𝒄〗_𝟏 𝒄_𝟐)/(√(〖𝒂_𝟏〗^𝟐 + 〖𝒃_𝟏〗^𝟐+ 〖𝒄_𝟏〗^𝟐 ) √(〖𝒂_𝟐〗^𝟐 +〖〖 𝒃〗_𝟐〗^𝟐+ 〖𝒄_𝟐〗^𝟐 ))| Angle between the pair of lines (𝑥 − 𝑥1)/𝑎1 = (𝑦 − 𝑦1)/𝑏1 = (𝑧 − 𝑧1)/𝑐1 and (𝑥 − 𝑥2)/𝑎2 = (𝑦 − 𝑦2)/𝑏2 = (𝑧 − 𝑧2)/𝑐2 is given by cos θ = |(𝒂_𝟏 𝒂_𝟐 + 𝒃_𝟏 𝒃_𝟐 +〖 𝒄〗_𝟏 𝒄_𝟐)/(√(〖𝒂_𝟏〗^𝟐 + 〖𝒃_𝟏〗^𝟐+ 〖𝒄_𝟏〗^𝟐 ) √(〖𝒂_𝟐〗^𝟐 +〖〖 𝒃〗_𝟐〗^𝟐+ 〖𝒄_𝟐〗^𝟐 ))| (𝒙 − 𝟐)/𝟐 = (𝒚 − 𝟏)/𝟓 = (𝒛 + 𝟑)/( − 𝟑) (𝑥 − 2)/2 = (𝑦 − 1)/5 = (𝑧 − (−3))/( − 3) Comparing with (𝑥 − 𝑥1)/𝑎1 = (𝑦 − 𝑦1)/𝑏1 = (𝑧 − 𝑧1)/𝑐1 x1 = 2, y1 = 1, z1 = –3 & 𝑎1 = 2, b1 = 5, c1 = –3 (𝒙 + 𝟐)/( − 𝟏) = (𝒚 − 𝟒)/𝟖 = (𝒛 − 𝟓)/𝟒 (𝑥 − (− 2))/( − 1) = (𝑦 − 4)/8 = (𝑧 − 5)/4 Comparing with (𝑥 − 𝑥2)/𝑎2 = (𝑦 − 𝑦2)/𝑏2 = (𝑧 − 𝑧2)/𝑐2 𝑥2 = − 2, y2 = 4, z2 = 5 & 𝑎2 = –1, 𝑏2 = 8, 𝑐2 = 4 Now, cos θ = |(𝒂_𝟏 𝒂_𝟐 + 𝒃_𝟏 𝒃_𝟐 +〖 𝒄〗_𝟏 𝒄_𝟐)/(√(〖𝒂_𝟏〗^𝟐 + 〖𝒃_𝟏〗^𝟐+ 〖𝒄_𝟏〗^𝟐 ) √(〖𝒂_𝟐〗^𝟐 +〖〖 𝒃〗_𝟐〗^𝟐+ 〖𝒄_𝟐〗^𝟐 ))| = |((2 × −1) + (5 × 8) + ( − 3 × 4) )/(√(2^2 + 5^2 + 〖(−3)〗^2 ) √(〖(−1)〗^2 + 8^2 + 4^2 ))| = |( −2 + 40 + (−12) )/(√(4 + 25 + 9) √(1 + 64 + 16))| = |26/(√38 √81)| = |26/(√38 × 9)| = 26/(9√38 ) So, cos θ = 26/(9√38 ) ∴ θ = cos−1 (𝟐𝟔/(𝟗√𝟑𝟖 )) Therefore, the angle between the given lines is cos-1 (26/(9√38 )). = |( −2 + 40 + (−12) )/(√(4 + 25 + 9) √(1 + 64 + 16))| = |26/(√38 √81)| = |26/(√38 × 9)| = 26/(9√38 ) So, cos θ = 26/(9√38 ) ∴ θ = cos−1 (𝟐𝟔/(𝟗√𝟑𝟖 )) Therefore, the angle between the given lines is cos-1 (26/(9√38 )).

Ask a doubt (live)
Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.