Examples

Chapter 11 Class 12 Three Dimensional Geometry
Serial order wise

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Example 23 Find the angle between the two planes 3x β 6y + 2z = 7 and 2x + 2y β 2z =5.Angle between two planes A1x + B1y + C1z = d1 and A2x + B2y + C2z = d2 is given by cos ΞΈ = |(π¨_π π¨_π + π©_π π©_π + πͺ_π πͺ_π)/(β(γπ¨_πγ^π + γπ©_πγ^π + γπͺ_πγ^π ) β(γπ¨_πγ^π + γπ©_πγ^π + γπͺ_πγ^π ))| Given the two planes are 3x β 6y + 2z = 7 Comparing with A1x + B1y + C1z = d1 A1 = 3 , B1 = β6 , C1 = 2 , π_1= 7 2x + 2y β 2z = 5 Comparing with A2x + B2y + C2z = d2 A2 = 2 , B2 = 2 , C2 = β2 , π_2= 5 So, cos ΞΈ = |((3 Γ 2) + (β6 Γ 2) + (2 Γ β2))/(β(3^2 + γ(β6)γ^2 + 2^2 ) β(2^2 + 2^2 + γ(β2)γ^2 ))| = |(6 + (β12) + (β4))/(β(9 + 36 + 4) Γβ(4 + 4 + 4))| = |(β10)/(β(49 ) Γβ12)| = |(β10)/(7 Γβ(4Γ3))| = 10/(7 Γ 2 Γ β3) = 5/(7β3) = 5/(7β3) Γ β3/β3 = (5β3)/21 So, cos ΞΈ = (5β3)/21 β΄ ΞΈ = γπππγ^(βπ) ((πβπ)/ππ) Therefore, the angle between the two planes is γπππ γ^(β1) ((5β3)/21) E