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

Examples 23 last slide.jpg

 


  1. Class 12
  2. Important Question for exams Class 12

Transcript

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)

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
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.