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

Examples 23 last slide.jpg

 


  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)

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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.