Ex 11.3, 6 - Find equations of planes passing three points - Equation of plane - Passing Through 3 Non Collinear Points

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  1. Chapter 11 Class 12 Three Dimensional Geometry
  2. Serial order wise
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Ex 11.3, 6 (Introduction) Find the equations of the planes that passes through three points. (a) (1, 1, – 1), (6, 4, – 5), (– 4, – 2, 3) Vector equation of a plane passing through three points with position vectors 𝑎īˇ¯, 𝑏īˇ¯, 𝑐īˇ¯ is ( rīˇ¯ − 𝑎īˇ¯) . ( 𝑏īˇ¯âˆ’ 𝑎īˇ¯)×( 𝑐īˇ¯âˆ’ 𝑎īˇ¯)īˇ¯ = 0 Ex 11.3, 6 Find the equations of the planes that passes through three points. (a) (1, 1, – 1), (6, 4, – 5), (– 4, – 2, 3) Vector equation of a plane passing through three points with position vectors 𝑎īˇ¯, 𝑏īˇ¯, 𝑐īˇ¯ is ( rīˇ¯ − 𝒂īˇ¯) . ( 𝒃īˇ¯âˆ’ 𝒂īˇ¯)×( 𝒄īˇ¯âˆ’ 𝒂īˇ¯)īˇ¯ = 0 Now, the plane passes through the points ( 𝒃īˇ¯ − 𝒂īˇ¯) = (6 𝑖īˇ¯ + 4 𝑗īˇ¯ – 5 𝑘īˇ¯) − (1 𝑖īˇ¯ + 1 𝑗īˇ¯ − 1 𝑘īˇ¯) = (6 −1) 𝑖īˇ¯ + (4 − 1) 𝑗īˇ¯ + (−5 − (− 1)) 𝑘īˇ¯ = 5 𝒊īˇ¯ + 3 𝒋īˇ¯ − 4 𝒌īˇ¯ ( 𝒄īˇ¯ − 𝒂īˇ¯) = (− 4 𝑖īˇ¯ − 2 𝑗īˇ¯ + 3 𝑘īˇ¯) − (1 𝑖īˇ¯ + 1 𝑗īˇ¯ − 1 𝑘īˇ¯) = (−4 − 1) 𝑖īˇ¯ +(−2 −1) 𝑗īˇ¯ + (3 − (− 1)) 𝑘īˇ¯ = −5 𝒊īˇ¯ − 3 𝒋īˇ¯ + 4 𝒌īˇ¯ ( 𝒃īˇ¯ − 𝒂īˇ¯) × ( 𝒄īˇ¯ − 𝒂īˇ¯) = 𝑖īˇ¯īˇŽ 𝑗īˇ¯īˇŽ 𝑘īˇ¯īˇŽ5īˇŽ3īˇŽ − 4īˇŽ − 5īˇŽ − 3īˇŽ4īˇ¯īˇ¯ = – 𝑖īˇ¯īˇŽ 𝑗īˇ¯īˇŽ 𝑘īˇ¯īˇŽ5īˇŽ3īˇŽ − 4īˇŽ 5īˇŽ 3īˇŽâˆ’ 4īˇ¯īˇ¯ = 𝟎īˇ¯ This implies, the three points are collinear. Ex 11.3, 6 Find the equations of the planes that passes through three points. (b) (1, 1, 0), (1, 2, 1), (– 2, 2, – 1) Vector equation of a plane passing through three points with position vectors 𝑎īˇ¯, 𝑏īˇ¯, 𝑐īˇ¯ is ( rīˇ¯ − 𝒂īˇ¯) . ( 𝒃īˇ¯âˆ’ 𝒂īˇ¯)×( 𝒄īˇ¯ − 𝒂īˇ¯)īˇ¯ = 0 Now, the plane passes through the points ( 𝒄īˇ¯ − 𝒂īˇ¯) = (–2 𝑖īˇ¯ + 2 𝑗īˇ¯ + 1 𝑘īˇ¯) − (1 𝑖īˇ¯ + 1 𝑗īˇ¯ +0 𝑘īˇ¯) = (−2 − 1) 𝑖īˇ¯ + (2 − 1) 𝑗īˇ¯ + (−1 − 0) 𝑘īˇ¯ = −3 𝒊īˇ¯ + 1 𝒋īˇ¯ − 1 𝒌īˇ¯ ( 𝒃īˇ¯ − 𝒂īˇ¯) × ( 𝒄īˇ¯ − 𝒂īˇ¯) = 𝑖īˇ¯īˇŽ 𝑗īˇ¯īˇŽ 𝑘īˇ¯īˇŽ0īˇŽ1īˇŽ1īˇŽ − 3īˇŽ1īˇŽ − 1īˇ¯īˇ¯ = 𝑖īˇ¯ 1 ×−1īˇ¯âˆ’(1×1)īˇ¯ – 𝑗īˇ¯ 0 ×−1īˇ¯âˆ’(−3 ×1)īˇ¯ + 𝑘īˇ¯ 0 ×1īˇ¯âˆ’(−3 ×1)īˇ¯ = 𝑖īˇ¯(–1 – 1) – 𝑗īˇ¯ (0 + 3) + 𝑘īˇ¯ ( 0 + 3) = –2 𝒊īˇ¯ – 3 𝒋īˇ¯ + 3 𝒌īˇ¯ ∴ Vector equation of plane is 𝑟īˇ¯âˆ’ 1 𝑖īˇ¯+1 𝑗īˇ¯+0 𝑘īˇ¯īˇ¯ īˇ¯. −2 𝑖īˇ¯âˆ’3 𝑗īˇ¯+3 𝑘īˇ¯īˇ¯ = 0 𝒓īˇ¯âˆ’ 𝒊īˇ¯+ 𝒋īˇ¯īˇ¯ īˇ¯. −𝟐 𝒊īˇ¯âˆ’đŸ‘ 𝒋īˇ¯+𝟑 𝒌īˇ¯īˇ¯ = 𝟎 Finding Cartesian equation Put 𝒓īˇ¯ = x 𝒊īˇ¯ + y 𝒋īˇ¯ + z 𝒌īˇ¯ 𝑟īˇ¯âˆ’ 𝑖īˇ¯+ 𝑗īˇ¯īˇ¯ īˇ¯. −2 𝑖īˇ¯âˆ’3 𝑗īˇ¯+3 𝑘īˇ¯īˇ¯ = 0 đ‘Ĩ 𝑖īˇ¯+đ‘Ļ 𝑗īˇ¯+𝑧 𝑘īˇ¯īˇ¯âˆ’( 𝑖īˇ¯+ 𝑗īˇ¯)īˇ¯. −2 𝑖īˇ¯âˆ’3 𝑗īˇ¯+3 𝑘īˇ¯īˇ¯ = 0 đ‘Ĩ−1īˇ¯ 𝑖īˇ¯ + đ‘Ļ−1īˇ¯ 𝑗īˇ¯+𝑧 𝑘īˇ¯īˇ¯. −2 𝑖īˇ¯âˆ’3 𝑗īˇ¯+3 𝑘īˇ¯īˇ¯ = 0 –2(x − 1) + (−3)(y − 1) + 3(z) = 0 –2x + 2 − 3y + 3 + 3z = 0 2x + 3y – 3z = 5 ∴ Equation of plane in Cartesian form is 2x + 3y – 3z = 5

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