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Chapter 11 Class 12 Three Dimensional Geometry

Serial order wise

Last updated at March 22, 2023 by Teachoo

Example 17 (introduction) Find the vector and cartesian equations of the plane which passes through the point (5, 2, – 4) and perpendicular to the line with direction ratios 2, 3, – 1.Vector equation of a plane passing through a point (x1, y1, z1) and perpendicular to a line with direction ratios A, B, C is [𝑟 ⃗ −(𝑥1𝑖 ̂ + 𝑦1𝑗 ̂ + 𝑧1𝑘 ̂)]. (A𝑖 ̂ + B𝑗 ̂ + C𝑘 ̂) = 0 or (𝑟 ⃗ − 𝑎 ⃗).𝑛 ⃗ = 0 ("A" 𝑃) ⃗ is perpendicular to "n" ⃗ So, ("A" P) ⃗ . "n" ⃗ = 0 ("r" ⃗ − "a" ⃗)."n" ⃗ = 0 Example 17 Find the vector and Cartesian equations of the plane which passes through the point (5, 2, – 4) and perpendicular to the line with direction ratios 2, 3, – 1.Vector form Equation of plane passing through point A whose position vector is 𝒂 ⃗ & perpendicular to 𝒏 ⃗ is (𝒓 ⃗ − 𝒂 ⃗) . 𝒏 ⃗ = 0 Given Plane passes through (5, 2, −4) So 𝒂 ⃗ = 5𝑖 ̂ + 2𝑗 ̂ − 4𝑘 ̂ Direction ratios of line perpendicular to plane = 2, 3, −1 So, "n" ⃗ = 2𝑖 ̂ + 3𝑗 ̂ − 1𝑘 ̂ Equation of plane in vector form is (𝑟 ⃗ − 𝑎 ⃗) . 𝑛 ⃗ = 0 [𝒓 ⃗−(𝟓𝒊 ̂+𝟐𝒋 ̂−𝟒𝒌 ̂)]. (𝟐𝒊 ̂+𝟑𝒋 ̂−𝒌 ̂) = 0 Now, Finding Cartesian form using two methods Cartesian form (Method 1) Vector equation is [𝑟 ⃗−(5𝑖 ̂+2𝑗 ̂−4𝑘 ̂)]. (2𝑖 ̂+3𝑗 ̂−𝑘 ̂) = 0 Put 𝒓 ⃗ = x𝒊 ̂ + y𝒋 ̂ + z𝒌 ̂ [(𝑥𝑖 ̂+𝑦𝑗 ̂+𝑧𝑘 ̂ )−(5𝑖 ̂+2𝑗 ̂−4𝑘 ̂)].(2𝑖 ̂ + 3𝑗 ̂ − 𝑘 ̂) = 0 [(𝑥−5) 𝑖 ̂+(𝑦−2) 𝑗 ̂+ (𝑧−(− 4))𝑘 ̂ ].(2𝑖 ̂ + 3𝑗 ̂ − 𝑘 ̂) = 0 2(x − 5) + 3 (y − 2) + (− 1)(z + 4) = 0 2x − 10 + 3y − 6 − z − 4 = 0 2x + 3y − z − 20 = 0 2x + 3y − z = 20 Therefore equation of plane in Cartesian form is 2x + 3y − z = 20 Cartesian form (Method 2) Equation of plane passing through (x1, y1, z1) and perpendicular to a line with direction ratios A, B, C is A(x − x1) + B(y − y1) + c (z − z1) = 0 Since the plane passes through (5, 2, −4) x1 = 5, y1 = 2, z1 = −4 Direction ratios of line perpendicular to plane = 2, 3, −1 ∴ A = 2, B = 3, C = −1 Therefore, equation of line in Cartesian form is 2(x − 5) + 3 (y − 2) + (−1) (x − (−4)) = 0 2 (x − 5) + 3(y − 2) − 1 (z + 4) = 0 2x − 10 + 3y − 6 − z − 4 = 0 2x + 3y − z − 20 = 0 2x + 3y − z = 20 Therefore equation of plane in Cartesian form is 2x + 3y − z = 20