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Ex 11.2

Ex 11.2, 1

Ex 11.2, 2

Ex 11.2, 3 Important

Ex 11.2, 4

Ex 11.2, 5 Important

Ex 11.2, 6

Ex 11.2, 7 Important

Ex 11.2, 8

Ex 11.2, 9 Important

Ex 11.2, 10 (i) Important

Ex 11.2, 10 (ii)

Ex 11.2, 11 (i) Important You are here

Ex 11.2, 11 (ii)

Ex 11.2, 12 Important

Ex 11.2, 13

Ex 11.2, 14 Important

Ex 11.2, 15 Important

Ex 11.2, 16

Ex 11.2, 17 Important

Chapter 11 Class 12 Three Dimensional Geometry

Serial order wise

Last updated at Aug. 24, 2021 by Teachoo

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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 )).