1. Chapter 11 Class 12 Three Dimensional Geometry
2. Serial order wise

Transcript

Example, 6 Find the vector and the Cartesian equations of the line through the point (5, 2, โ 4) and which is parallel to the vector 3๐ย ฬ + 2๐ย ฬ โ 8๐ย ฬ . Vector equation Equation of a line passing through a point with position vector ๐ย โ , and parallel to a vector ๐ย โ is ๐ย โ = ๐ย โ + ๐๐ย โ Since line passes through (5, 2, โ 4) ๐ย โ = 5๐ย ฬ + 2๐ย ฬ โ 4๐ย ฬ Since line is parallel to 3๐ย ฬ + 2๐ย ฬ โ 8๐ย ฬ ๐ย โ = 3๐ย ฬ + 2๐ย ฬ โ 8๐ย ฬ Equation of line ๐ย โ = ๐ย โ + ๐๐ย โ ๐ย โ = (5๐ย ฬ + 2๐ย ฬ โ 4๐ย ฬ) + ๐ (3๐ย ฬ + 2๐ย ฬ โ 8๐ย ฬ) Therefore, equation of line in vector form is ๐ย โ = (5๐ย ฬ + 2๐ย ฬ โ 4๐ย ฬ) + ๐ (3๐ย ฬ + 2๐ย ฬ โ 8๐ย ฬ) Cartesian equation Equation of a line passing through a point (x, y, z) and parallel to a line with direction ratios a, b, c is (๐ฅ โ ๐ฅ1)/๐ = (๐ฆ โ ๐ฆ1)/๐ = (๐ง โ ๐ง1)/๐ Since line passes through (5, 2, โ4) ๐ฅ1 = 5, y1 = 2 , z1 = โ4 Also, line is parallel to 3๐ย ฬ + 2๐ย ฬ โ 8๐ย ฬ , ๐ = 3, b = 2, c = โ 8 Equation of line in Cartesian form is (๐ฅ โ ๐ฅ1)/๐ = (๐ฆ โ ๐ฆ1)/๐ = (๐ง โ ๐ง1)/๐ (๐ฅ โ 5)/3 = (๐ฆ โ 2)/2 = (๐ง โ ( โ 4))/( โ 8) (๐ โ ๐)/๐ = (๐ โ ๐)/๐ = (๐ + ๐)/(โ๐)

Serial order wise