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

Transcript

Example 23 Find the angle between the two planes 3x โ€“ 6y + 2z = 7 and 2x + 2y โ€“ 2z =5.Angle between two planes A1x + B1y + C1z = d1 and A2x + B2y + C2z = d2 is given by cos ฮธ = |(๐‘จ_๐Ÿ ๐‘จ_๐Ÿ + ๐‘ฉ_๐Ÿ ๐‘ฉ_๐Ÿ + ๐‘ช_๐Ÿ ๐‘ช_๐Ÿ)/(โˆš(ใ€–๐‘จ_๐Ÿใ€—^๐Ÿ + ใ€–๐‘ฉ_๐Ÿใ€—^๐Ÿ + ใ€–๐‘ช_๐Ÿใ€—^๐Ÿ ) โˆš(ใ€–๐‘จ_๐Ÿใ€—^๐Ÿ + ใ€–๐‘ฉ_๐Ÿใ€—^๐Ÿ + ใ€–๐‘ช_๐Ÿใ€—^๐Ÿ ))| Given the two planes are 3x โˆ’ 6y + 2z = 7 Comparing with A1x + B1y + C1z = d1 A1 = 3 , B1 = โ€“6 , C1 = 2 , ๐‘‘_1= 7 2x + 2y โˆ’ 2z = 5 Comparing with A2x + B2y + C2z = d2 A2 = 2 , B2 = 2 , C2 = โ€“2 , ๐‘‘_2= 5 So, cos ฮธ = |((3 ร— 2) + (โˆ’6 ร— 2) + (2 ร— โˆ’2))/(โˆš(3^2 + ใ€–(โˆ’6)ใ€—^2 + 2^2 ) โˆš(2^2 + 2^2 + ใ€–(โˆ’2)ใ€—^2 ))| = |(6 + (โˆ’12) + (โˆ’4))/(โˆš(9 + 36 + 4) ร—โˆš(4 + 4 + 4))| = |(โˆ’10)/(โˆš(49 ) ร—โˆš12)| = |(โˆ’10)/(7 ร—โˆš(4ร—3))| = 10/(7 ร— 2 ร— โˆš3) = 5/(7โˆš3) = 5/(7โˆš3) ร— โˆš3/โˆš3 = (5โˆš3)/21 So, cos ฮธ = (5โˆš3)/21 โˆด ฮธ = ใ€–๐’„๐’๐’”ใ€—^(โˆ’๐Ÿ) ((๐Ÿ“โˆš๐Ÿ‘)/๐Ÿ๐Ÿ) Therefore, the angle between the two planes is ใ€–๐‘๐‘œ๐‘ ใ€—^(โˆ’1) ((5โˆš3)/21) E

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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.