Example 7 - Find vector equation for line (-1, 0, 2), (3, 4, 6) - Equation of line  - given 2 points

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  1. Chapter 11 Class 12 Three Dimensional Geometry
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Example, 7 Find the vector equation for the line passing through the points (–1, 0, 2) and (3, 4, 6). Vector equation Vector equation of a line passing though two points with position vectors 𝑎﷯ and 𝑏﷯ is 𝑟﷯ = 𝑎 ﷯ + 𝜆 ( 𝑏﷯ − 𝑎﷯) Given, Let two points be A (–1, 0, 2) & B(3, 4, 6) So, 𝑟﷯ = ( − 𝑖﷯ + 0 𝑗﷯ + 2 𝑘﷯) + 𝜆 3 𝑖﷯+4𝑗+6 𝑘﷯﷯ − (− 𝑖﷯ + 0 𝑗﷯ + 2 𝑘﷯)﷯ = – 𝑖﷯ + 0 𝑗﷯ + 2 𝑘﷯ + 𝜆 3 𝑖﷯ + 4𝑗 + 6 𝑘﷯ + 𝑖﷯ − 0 𝑗﷯ − 2 𝑘﷯﷯ = − 𝑖﷯ + 0 𝑗﷯ + 2 𝑘﷯ + 𝜆 3 + 1﷯ 𝑖﷯ + 4 − 0﷯ 𝑗﷯ + (6 − 2) 𝑘﷯﷯ = − 𝑖﷯ + 0 𝑗﷯ + 2 𝑘﷯ + 𝜆 4 𝑖﷯ + 4 𝑗﷯ + 4 𝑘﷯﷯ = − 𝒊﷯ + 2 𝒌﷯ + 𝜆 𝟒 𝒊﷯ + 𝟒 𝒋﷯ + 𝟒 𝒌﷯﷯ Therefore, the vector equation is − 𝑖﷯ + 2 𝑘﷯ + 𝜆 4 𝑖﷯ + 4 𝑗﷯ + 4 𝑘﷯﷯

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