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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Ex 11.2, 9 Find the angle between the following pairs of lines: (i) (𝑥 − 2)/2 = (𝑦 − 1)/5 = (𝑧 + 3)/(−3) and (𝑥 + 2)/(−1) = (𝑦 − 4)/8 = (𝑧 − 5)/4Angle 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 )).

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.