Ex 11.2, 7 - Chapter 11 Class 12 Three Dimensional Geometry

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Ex 11.2, 7 The Cartesian equation of a line is (𝑥 − 5)/3 = (𝑦 + 4)/7 = (𝑧 − 6)/2. Write its vector form.Cartesian equation : (𝑥 − 5)/3 = (𝑦 + 4)/7 = (𝑧 − 6)/2 (𝑥 − 5)/3 = (𝑦 − (−4))/7 = (𝑧 − 6)/2 Equation of a line in Cartesian form is given by (𝑥 − 𝑥1)/𝑎 = (𝑦 − 𝑦1)/𝑏 = (𝑧 − 𝑧1)/𝑐 Comparing (1) and (2), 𝑥1 = 5, 𝑦1 = −4, 𝑧1 = 6 𝑎 = 3, b = 7, c = 2 Cartesian equation : (𝑥 − 5)/3 = (𝑦 + 4)/7 = (𝑧 − 6)/2 (𝑥 − 5)/3 = (𝑦 − (−4))/7 = (𝑧 − 6)/2 Equation of a line in Cartesian form is given by (𝑥 − 𝑥1)/𝑎 = (𝑦 − 𝑦1)/𝑏 = (𝑧 − 𝑧1)/𝑐 Comparing (1) and (2), 𝑥1 = 5, 𝑦1 = −4, 𝑧1 = 6 𝑎 = 3, b = 7, c = 2 Equation of line in vector form is 𝑟 ⃗ = 𝑎 ⃗ + 𝜆𝑏 ⃗ Where 𝑎 ⃗ = 𝑥1𝑖 ̂ + y1𝑗 ̂ + z1𝑘 ̂ = 5𝑖 ̂ − 4𝑗 ̂ + 6𝑘 ̂ & 𝑏 ⃗ = 𝑎𝑖 ̂ + b𝑗 ̂ + c𝑘 ̂ = 3𝑖 ̂ + 7𝑗 ̂ + 2𝑘 ̂ Now, 𝑟 ⃗ = (5𝒊 ̂ − 4𝒋 ̂ + 6𝒌 ̂) + 𝜆 (3𝒊 ̂ + 7𝒋 ̂ + 2𝒌 ̂)