Example 25 - Find angle between line and plane - Class 12

Example, 25 - Chapter 11 Class 12 Three Dimensional Geometry - Part 2
Example, 25 - Chapter 11 Class 12 Three Dimensional Geometry - Part 3

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Question 15 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 has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.