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  1. Chapter 11 Class 12 Three Dimensional Geometry
  2. Serial order wise

Transcript

Example, 25 Find the angle between the line (π‘₯ + 1)/2 = 𝑦/3 = (𝑧 βˆ’ 3)/6 And the plane 10x + 2y – 11z = 3. The angle between a line (π‘₯ βˆ’ π‘₯_1)/π‘Ž = (𝑦 βˆ’ 𝑦_1)/𝑏 = (𝑧 βˆ’γ€– 𝑧〗_1)/𝑐 and the normal to the plane Ax + By + Cz = D is given by cos ΞΈ = |(π΄π‘Ž + 𝐡𝑏 + 𝐢𝑐)/(√(π‘Ž^2 + 𝑏^2 +γ€– 𝑐〗^2 ) √(𝐴^2 +γ€– 𝐡〗^2 +γ€– 𝐢〗^2 ))| So, angle between line and plane is given by sin 𝝓 = |(π΄π‘Ž + 𝐡𝑏 + 𝐢𝑐)/(√(π‘Ž^2 + 𝑏^2 + 𝑐^2 )+√(𝐴^2 + 𝐡^2 +γ€– 𝐢〗^2 ))| Given, the line is (π‘₯ + 1)/2 = 𝑦/3 = (𝑧 βˆ’ 3)/6 (π‘₯ βˆ’ (βˆ’1))/2 = (𝑦 βˆ’ 0)/3 = (𝑧 βˆ’ 3)/6 Comparing with (π‘₯ βˆ’γ€– π‘₯γ€—_1)/π‘Ž = (𝑦 βˆ’ 𝑦_1)/𝑏 = (𝑧 βˆ’ 𝑧_1)/𝑐 , π‘Ž = 2, b = 3, c = 6 The plane is 10x + 2y βˆ’ 11z = 3 Comparing with Ax + By + Cz = D, A = 10, B = 2, C = βˆ’11 So, sin Ο• = |((10 Γ— 2) + (2 Γ— 3) + (βˆ’11 Γ— 6))/(√(2^2 + 3^2 + 6^2 ) √(γ€–10γ€—^(2 )+γ€– 2γ€—^2 + γ€–(βˆ’11)γ€—^2 ))| = |(20 + 6 βˆ’ 66)/(√(4 + 9 + 36) √(100 + 4 + 121))| = |(βˆ’40)/(7 Γ— 15)| = 8/21 So, sin Ο• = 8/21 ∴ 𝝓 = γ€–π’”π’Šπ’γ€—^(βˆ’πŸ)⁑(πŸ–/𝟐𝟏) Therefore, the angle between the given line and plane is sin^(βˆ’1)⁑(8/21).

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.