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Example 8 - The Cartesian equation of a line is. Find vector - Equation of line  - given point and //vector

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  1. Chapter 11 Class 12th Three Dimensional Geometry
  2. Serial order wise
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Example 8 The Cartesian equation of a line is 𝑥 + 3﷮2﷯ = 𝑦 − 5﷮4﷯ = 𝑧 + 6﷮2﷯ Find the vector equation for the line. Cartesian equation : 𝑥 + 3﷮2﷯ = 𝑦 − 5﷮4﷯ = 𝑧 + 6﷮2﷯ 𝑥 − (− 3)﷮2﷯ = 𝑦 − 5﷮4﷯ = 𝑧 − (− 6)﷮2﷯ Equation of a line in Cartesian form is 𝑥 − 𝑥1﷮𝑎﷯ = 𝑦 − 𝑦1﷮𝑏﷯ = 𝑧 − 𝑧1﷮𝑐﷯ Comparing (1) and (2), 𝑥1= −3, 𝑦1= 5, 𝑧1= −6 & 𝑎 = 2, 𝑏 = 4, c = 2 Equation of line in vector form is 𝑟﷯ = 𝑎﷯ + 𝜆 𝑏﷯ where 𝑎﷯ = 𝑥1 𝑖﷯ + y1 𝑗﷯ + z1 𝑘﷯ = − 3 𝑖﷯ + 5 𝑗﷯ − 6 𝑘﷯ & 𝑏﷯ = 𝑎 𝑖﷯ + b 𝑗﷯ + c 𝑘﷯ = 2 𝑖﷯ + 4 𝑗﷯ + 2 𝑘﷯ Now, 𝑟﷯ = ( −3 𝒊﷯ + 5 𝒋﷯ − 6 𝒌﷯) + 𝜆 (2 𝒊﷯ + 4 𝒋﷯ + 2 𝒌﷯) Therefore, equation of line in vector form is ( −3 𝑖﷯ + 5 𝑗﷯ − 6 𝑘﷯) + 𝜆 (2 𝑖﷯ + 4 𝑗﷯ + 2 𝑘﷯)

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