Example 22
Last updated at Dec. 8, 2016 by Teachoo
Transcript
Example 22 Find the angle between the two planes 2x + y โ 2z = 5 and 3x โ 6y โ 2z = 7 using vector method. Angle between two planes ๐ย โ. (๐1)ย โ = d1 and ๐ย โ.(๐2)ย โ = d2 is given by cos ฮธ = |((๐๐)ย โ . (๐๐)ย โ)/|(๐๐)ย โ ||(๐๐)ย โ | | Given, the two planes are 2x + y โ 2z = 5 Comparing with A1x + B1y + C1z = d1 Direction ratios of normal = 2, 1, โ2 (๐1)ย โ = 2๐ย ฬ + 1๐ย ฬ โ 2๐ย ฬ Magnitude of (๐1)ย โ = โ(22+12+( โ 2)2) |(๐1)ย โ |= โ(4+1+4) = โ9 = 3 3x โ 6y โ 2z = 7 Comparing with A2x + B2y + C2z = d2 Direction ratios of normal = 3,โ6, โ2 (๐2)ย โ = 3๐ย ฬ โ 6๐ย ฬ โ 2๐ย ฬ Magnitude of (๐2)ย โ = โ(32+(โ 6)2+( โ 2)2) |(๐2)ย โ |= โ(9+36+4) = โ49 = 7 So, cos ฮธ = |((2๐ย ฬ" " + 1๐ย ฬ" " โ 2๐ย ฬ ) . (3๐ย ฬ" " โ 6๐ย ฬ" " โ 2๐ย ฬ ))/(3 ร 7)| = |((2 ร 3) + (1 ร โ6) + (โ2 ร โ2))/21| = |(6 โ 6 + 4)/21| = 4/21 So, cos ฮธ = 4/21 โด ฮธ = cos-1(๐/๐๐) Therefore, two angle between the two planes is cos-1(4/21)
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About the Author
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.