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Example 23 - Find angle between 3x - 6y + 2z = 7 and 2x + 2y

Example 23 - Chapter 11 Class 12 Three Dimensional Geometry - Part 2
Example 23 - Chapter 11 Class 12 Three Dimensional Geometry - Part 3

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Example 23 Find the angle between the two planes 3x – 6y + 2z = 7 and 2x + 2y – 2z =5.Angle between two planes A1x + B1y + C1z = d1 and A2x + B2y + C2z = d2 is given by cos ΞΈ = |(𝑨_𝟏 𝑨_𝟐 + 𝑩_𝟏 𝑩_𝟐 + π‘ͺ_𝟏 π‘ͺ_𝟐)/(√(〖𝑨_πŸγ€—^𝟐 + 〖𝑩_πŸγ€—^𝟐 + γ€–π‘ͺ_πŸγ€—^𝟐 ) √(〖𝑨_πŸγ€—^𝟐 + 〖𝑩_πŸγ€—^𝟐 + γ€–π‘ͺ_πŸγ€—^𝟐 ))| Given the two planes are 3x βˆ’ 6y + 2z = 7 Comparing with A1x + B1y + C1z = d1 A1 = 3 , B1 = –6 , C1 = 2 , 𝑑_1= 7 2x + 2y βˆ’ 2z = 5 Comparing with A2x + B2y + C2z = d2 A2 = 2 , B2 = 2 , C2 = –2 , 𝑑_2= 5 So, cos ΞΈ = |((3 Γ— 2) + (βˆ’6 Γ— 2) + (2 Γ— βˆ’2))/(√(3^2 + γ€–(βˆ’6)γ€—^2 + 2^2 ) √(2^2 + 2^2 + γ€–(βˆ’2)γ€—^2 ))| = |(6 + (βˆ’12) + (βˆ’4))/(√(9 + 36 + 4) Γ—βˆš(4 + 4 + 4))| = |(βˆ’10)/(√(49 ) Γ—βˆš12)| = |(βˆ’10)/(7 Γ—βˆš(4Γ—3))| = 10/(7 Γ— 2 Γ— √3) = 5/(7√3) = 5/(7√3) Γ— √3/√3 = (5√3)/21 So, cos ΞΈ = (5√3)/21 ∴ ΞΈ = 〖𝒄𝒐𝒔〗^(βˆ’πŸ) ((πŸ“βˆšπŸ‘)/𝟐𝟏) Therefore, the angle between the two planes is γ€–π‘π‘œπ‘ γ€—^(βˆ’1) ((5√3)/21) E

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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.