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Ex 11.3, 6 - Chapter 11 Class 12 Three Dimensional Geometry - Part 5

Ex 11.3, 6 - Chapter 11 Class 12 Three Dimensional Geometry - Part 6
Ex 11.3, 6 - Chapter 11 Class 12 Three Dimensional Geometry - Part 7

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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 π‘Ž βƒ—, 𝑏 βƒ—, 𝑐 βƒ— is ("r" βƒ— βˆ’ 𝒂 βƒ—) . [(𝒃 βƒ—βˆ’π’‚ βƒ—)Γ—(𝒄 βƒ— βˆ’π’‚ βƒ—)] = 0 Now, the plane passes through the points A (1, 1, 0) π‘Ž βƒ— = 1𝑖 Μ‚ + 1𝑗 Μ‚ + 0π‘˜ Μ‚ B (1, 2, 1) 𝑏 βƒ— = 1𝑖 Μ‚ + 2𝑗 Μ‚ + 1π‘˜ Μ‚ C (βˆ’2, 2, βˆ’1) 𝑐 βƒ— = βˆ’2𝑖 Μ‚ + 2𝑗 Μ‚ βˆ’ 1π‘˜ Μ‚ (𝒃 βƒ— βˆ’ 𝒂 βƒ—) = (1𝑖 Μ‚ + 2𝑗 Μ‚ + 1π‘˜ Μ‚) βˆ’ (1𝑖 Μ‚ + 1𝑗 Μ‚ + 0π‘˜ Μ‚) = (1 βˆ’ 1) 𝑖 Μ‚ + (2 βˆ’ 1)𝑗 Μ‚ + (1 βˆ’ 0) π‘˜ Μ‚ = 0π’Š Μ‚ + 1𝒋 Μ‚ + 1π’Œ Μ‚ (𝒄 βƒ— βˆ’ 𝒂 βƒ—) = (–2𝑖 Μ‚ + 2𝑗 Μ‚ + 1π‘˜ Μ‚) βˆ’ (1𝑖 Μ‚ + 1𝑗 Μ‚ +0π‘˜ Μ‚) = (βˆ’2 βˆ’ 1)𝑖 Μ‚ + (2 βˆ’ 1)𝑗 Μ‚ + (βˆ’1 βˆ’ 0) π‘˜ Μ‚ = βˆ’3π’Š Μ‚ + 1𝒋 Μ‚ βˆ’ 1π’Œ Μ‚ (𝒃 βƒ— βˆ’ 𝒂 βƒ—) Γ— (𝒄 βƒ— βˆ’ 𝒂 βƒ—) = |β– 8(𝑖 Μ‚&𝑗 Μ‚&π‘˜ Μ‚@0&1&1@ βˆ’ 3&1& βˆ’ 1)| = 𝑖 Μ‚ [(1Γ—βˆ’1)βˆ’(1Γ—1)] – 𝑗 Μ‚ [(0Γ—βˆ’1)βˆ’(βˆ’3 Γ—1)] + (π‘˜ ) Μ‚[(0Γ—1)βˆ’(βˆ’3 Γ—1)] = 𝑖 Μ‚(–1 – 1) – 𝑗 Μ‚ (0 + 3) + π‘˜ Μ‚ ( 0 + 3) = –2π’Š Μ‚ – 3𝒋 Μ‚ + 3π’Œ Μ‚ ∴ Vector equation of plane is [π‘Ÿ βƒ—βˆ’(1𝑖 Μ‚+1𝑗 Μ‚+0π‘˜ Μ‚ ) ].(βˆ’2𝑖 Μ‚βˆ’3𝑗 Μ‚+3π‘˜ Μ‚ ) = 0 [𝒓 βƒ—βˆ’(π’Š Μ‚+𝒋 Μ‚ ) ].(βˆ’πŸπ’Š Μ‚βˆ’πŸ‘π’‹ Μ‚+πŸ‘π’Œ Μ‚ ) = 𝟎 Finding Cartesian equation Put 𝒓 βƒ— = xπ’Š Μ‚ + y𝒋 Μ‚ + zπ’Œ Μ‚ [π‘Ÿ βƒ—βˆ’(𝑖 Μ‚+𝑗 Μ‚ ) ].(βˆ’2𝑖 Μ‚βˆ’3𝑗 Μ‚+3π‘˜ Μ‚ ) = 0 [(π‘₯𝑖 Μ‚+𝑦𝑗 Μ‚+π‘§π‘˜ Μ‚ )βˆ’(𝑖 Μ‚+𝑗 Μ‚)]. (βˆ’2𝑖 Μ‚βˆ’3𝑗 Μ‚+3π‘˜ Μ‚ ) = 0 [(π‘₯βˆ’1) 𝑖 Μ‚ +(π‘¦βˆ’1) 𝑗 Μ‚+π‘§π‘˜ Μ‚ ]. (βˆ’2𝑖 Μ‚βˆ’3𝑗 Μ‚+3π‘˜ Μ‚ ) = 0 –2(x βˆ’ 1) + (βˆ’3)(y βˆ’ 1) + 3(z) = 0 –2x + 2 βˆ’ 3y + 3 + 3z = 0 2x + 3y – 3z = 5 ∴ Equation of plane in Cartesian form is 2x + 3y – 3z = 5

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.