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

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 ๐จ_๐ + ๐ฉ_๐ ๐ฉ_๐ + ๐ช_๐ ๐ช_๐)/(โ(ใ๐จ_๐ใ^๐ + ใ๐ฉ_๐ใ^๐ + ใ๐ช_๐ใ^๐ ) โ(ใ๐จ_๐ใ^๐ + ใ๐ฉ_๐ใ^๐ + ใ๐ช_๐ใ^๐ ))| Given the two planes are 3x โ 6y + 2z = 7 Comparing with A1x + B1y + C1z = d1 A1 = 3 , B1 = โ6 , C1 = 2 , ๐_1= 7 2x + 2y โ 2z = 5 Comparing with A2x + B2y + C2z = d2 A2 = 2 , B2 = 2 , C2 = โ2 , ๐_2= 5 So, cos ฮธ = |((3 ร 2) + (โ6 ร 2) + (2 ร โ2))/(โ(3^2 + ใ(โ6)ใ^2 + 2^2 ) โ(2^2 + 2^2 + ใ(โ2)ใ^2 ))| = |(6 + (โ12) + (โ4))/(โ(9 + 36 + 4) รโ(4 + 4 + 4))| = |(โ10)/(โ(49 ) รโ12)| = |(โ10)/(7 รโ(4ร3))| = 10/(7 ร 2 ร โ3) = 5/(7โ3) = 5/(7โ3) ร โ3/โ3 = (5โ3)/21 So, cos ฮธ = (5โ3)/21 โด ฮธ = ใ๐๐๐ใ^(โ๐) ((๐โ๐)/๐๐) Therefore, the angle between the two planes is ใ๐๐๐ ใ^(โ1) ((5โ3)/21)

Examples