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Ex 11.3, 6 - Find equation of planes that passes through 3 points

Ex 11.3, 6 - Chapter 11 Class 12 Three Dimensional Geometry - Part 2

Ex 11.3, 6 - Chapter 11 Class 12 Three Dimensional Geometry - Part 3
Ex 11.3, 6 - Chapter 11 Class 12 Three Dimensional Geometry - Part 4

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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 π‘Ž βƒ—, 𝑏 βƒ—, 𝑐 βƒ— is ("r" βƒ— βˆ’ π‘Ž βƒ—) . [(𝑏 βƒ—βˆ’π‘Ž βƒ—)Γ—(𝑐 βƒ—βˆ’π‘Ž βƒ—)] = 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 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 (𝒃 βƒ— βˆ’ 𝒂 βƒ—) = (6𝑖 Μ‚ + 4𝑗 Μ‚ – 5π‘˜ Μ‚) βˆ’ (1𝑖 Μ‚ + 1𝑗 Μ‚ βˆ’ 1π‘˜ Μ‚) = (6 βˆ’1)𝑖 Μ‚ + (4 βˆ’ 1)𝑗 Μ‚ + (βˆ’5 βˆ’ (βˆ’1)) π‘˜ Μ‚ = 5π’Š Μ‚ + 3𝒋 Μ‚ βˆ’ 4π’Œ Μ‚ A (1, 1, βˆ’1) π‘Ž βƒ— = 1𝑖 Μ‚ + 1𝑗 Μ‚ βˆ’ 1π‘˜ Μ‚ B (6, 4, βˆ’5) 𝑏 βƒ— = 6𝑖 Μ‚ + 4𝑗 Μ‚ βˆ’ 5π‘˜ Μ‚ C ( βˆ’4, βˆ’2, 3) 𝑐 βƒ— = βˆ’4𝑖 Μ‚ βˆ’ 2𝑗 Μ‚ + 3π‘˜ Μ‚ (𝒄 βƒ— βˆ’ 𝒂 βƒ—) = (βˆ’4𝑖 Μ‚ βˆ’ 2𝑗 Μ‚ + 3π‘˜ Μ‚) βˆ’ (1𝑖 Μ‚ + 1𝑗 Μ‚ βˆ’ 1π‘˜ Μ‚) = (βˆ’4 βˆ’ 1)𝑖 Μ‚ +(βˆ’2 βˆ’ 1)𝑗 Μ‚ + (3 βˆ’ (βˆ’1)) π‘˜ Μ‚ = βˆ’5π’Š Μ‚ βˆ’ 3𝒋 Μ‚ + 4π’Œ Μ‚ (𝒃 βƒ— βˆ’ 𝒂 βƒ—) Γ— (𝒄 βƒ— βˆ’ 𝒂 βƒ—) = |β– 8(𝑖 Μ‚&𝑗 Μ‚&π‘˜ Μ‚@5&3&βˆ’4@βˆ’5&βˆ’3&4)| = – |β– 8(𝑖 Μ‚&𝑗 Μ‚&π‘˜ Μ‚@5&3&βˆ’4@ 5& 3&βˆ’4)| = 𝟎 βƒ— This implies, the three points are collinear. Using property: Since the two rows of the determinant are same, the value of determinant is zero. ∴ Vector equation of plane is [π‘Ÿ βƒ—βˆ’(𝑖 Μ‚+𝑗 Μ‚ βˆ’π‘˜ Μ‚ )] . 0 βƒ— = 0 Since, the above equation is satisfied for all values of π‘Ÿ βƒ—, Therefore, there will be infinite planes passing through the given 3 collinear points.

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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.