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Example 22 - Find angle between two planes using vector method - Angle between two planes

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  1. Chapter 11 Class 12 Three Dimensional Geometry
  2. Serial order wise
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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 ๐‘Ÿย โƒ—. (๐‘›1)ย โƒ— = d1 and ๐‘Ÿย โƒ—.(๐‘›2)ย โƒ— = d2 is given by cos ฮธ = |((๐’๐Ÿ)ย โƒ— . (๐’๐Ÿ)ย โƒ—)/|(๐’๐Ÿ)ย โƒ— ||(๐’๐Ÿ)ย โƒ— | | Given, the two planes are 2x + y โˆ’ 2z = 5 Comparing with A1x + B1y + C1z = d1 Direction ratios of normal = 2, 1, โˆ’2 (๐‘›1)ย โƒ— = 2๐‘–ย ฬ‚ + 1๐‘—ย ฬ‚ โˆ’ 2๐‘˜ย ฬ‚ Magnitude of (๐‘›1)ย โƒ— = โˆš(22+12+( โˆ’ 2)2) |(๐‘›1)ย โƒ— |= โˆš(4+1+4) = โˆš9 = 3 3x โ€“ 6y โˆ’ 2z = 7 Comparing with A2x + B2y + C2z = d2 Direction ratios of normal = 3,โˆ’6, โˆ’2 (๐‘›2)ย โƒ— = 3๐‘–ย ฬ‚ โˆ’ 6๐‘—ย ฬ‚ โˆ’ 2๐‘˜ย ฬ‚ Magnitude of (๐‘›2)ย โƒ— = โˆš(32+(โˆ’ 6)2+( โˆ’ 2)2) |(๐‘›2)ย โƒ— |= โˆš(9+36+4) = โˆš49 = 7 So, cos ฮธ = |((2๐‘–ย ฬ‚" " + 1๐‘—ย ฬ‚" " โˆ’ 2๐‘˜ย ฬ‚ ) . (3๐‘–ย ฬ‚" " โˆ’ 6๐‘—ย ฬ‚" " โˆ’ 2๐‘˜ย ฬ‚ ))/(3 ร— 7)| = |((2 ร— 3) + (1 ร— โˆ’6) + (โˆ’2 ร— โˆ’2))/21| = |(6 โˆ’ 6 + 4)/21| = 4/21 So, cos ฮธ = 4/21 โˆด ฮธ = cos-1(๐Ÿ’/๐Ÿ๐Ÿ) Therefore, two angle between the two planes is cos-1(4/21)

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