Plane

Chapter 11 Class 12 Three Dimensional Geometry
Serial order wise

### Transcript

Question 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 𝑎 ⃗, 𝑏 ⃗, 𝑐 ⃗ is ("r" ⃗ − 𝒂 ⃗) . [(𝒃 ⃗−𝒂 ⃗)×(𝒄 ⃗ −𝒂 ⃗)] = 0 Now, the plane passes through the points A (1, 1, 0) 𝑎 ⃗ = 1𝑖 ̂ + 1𝑗 ̂ + 0𝑘 ̂ B (1, 2, 1) 𝑏 ⃗ = 1𝑖 ̂ + 2𝑗 ̂ + 1𝑘 ̂ C (−2, 2, −1) 𝑐 ⃗ = −2𝑖 ̂ + 2𝑗 ̂ − 1𝑘 ̂ (𝒃 ⃗ − 𝒂 ⃗) = (1𝑖 ̂ + 2𝑗 ̂ + 1𝑘 ̂) − (1𝑖 ̂ + 1𝑗 ̂ + 0𝑘 ̂) = (1 − 1) 𝑖 ̂ + (2 − 1)𝑗 ̂ + (1 − 0) 𝑘 ̂ = 0𝒊 ̂ + 1𝒋 ̂ + 1𝒌 ̂ (𝒄 ⃗ − 𝒂 ⃗) = (–2𝑖 ̂ + 2𝑗 ̂ + 1𝑘 ̂) − (1𝑖 ̂ + 1𝑗 ̂ +0𝑘 ̂) = (−2 − 1)𝑖 ̂ + (2 − 1)𝑗 ̂ + (−1 − 0) 𝑘 ̂ = −3𝒊 ̂ + 1𝒋 ̂ − 1𝒌 ̂ (𝒃 ⃗ − 𝒂 ⃗) × (𝒄 ⃗ − 𝒂 ⃗) = |■8(𝑖 ̂&𝑗 ̂&𝑘 ̂@0&1&1@ − 3&1& − 1)| = 𝑖 ̂ [(1×−1)−(1×1)] – 𝑗 ̂ [(0×−1)−(−3 ×1)] + (𝑘 ) ̂[(0×1)−(−3 ×1)] = 𝑖 ̂(–1 – 1) – 𝑗 ̂ (0 + 3) + 𝑘 ̂ ( 0 + 3) = –2𝒊 ̂ – 3𝒋 ̂ + 3𝒌 ̂ ∴ Vector equation of plane is [𝑟 ⃗−(1𝑖 ̂+1𝑗 ̂+0𝑘 ̂ ) ].(−2𝑖 ̂−3𝑗 ̂+3𝑘 ̂ ) = 0 [𝒓 ⃗−(𝒊 ̂+𝒋 ̂ ) ].(−𝟐𝒊 ̂−𝟑𝒋 ̂+𝟑𝒌 ̂ ) = 𝟎 Finding Cartesian equation Put 𝒓 ⃗ = x𝒊 ̂ + y𝒋 ̂ + z𝒌 ̂ [𝑟 ⃗−(𝑖 ̂+𝑗 ̂ ) ].(−2𝑖 ̂−3𝑗 ̂+3𝑘 ̂ ) = 0 [(𝑥𝑖 ̂+𝑦𝑗 ̂+𝑧𝑘 ̂ )−(𝑖 ̂+𝑗 ̂)]. (−2𝑖 ̂−3𝑗 ̂+3𝑘 ̂ ) = 0 [(𝑥−1) 𝑖 ̂ +(𝑦−1) 𝑗 ̂+𝑧𝑘 ̂ ]. (−2𝑖 ̂−3𝑗 ̂+3𝑘 ̂ ) = 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

#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.