Last updated at May 29, 2018 by Teachoo

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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 ฬ + 2๐ ฬ โ 8๐ ฬ . Vector equation Equation of a line passing through a point with position vector ๐ โ , and parallel to a vector ๐ โ is ๐ โ = ๐ โ + ๐๐ โ Since line passes through (5, 2, โ 4) ๐ โ = 5๐ ฬ + 2๐ ฬ โ 4๐ ฬ Since line is parallel to 3๐ ฬ + 2๐ ฬ โ 8๐ ฬ ๐ โ = 3๐ ฬ + 2๐ ฬ โ 8๐ ฬ Equation of line ๐ โ = ๐ โ + ๐๐ โ ๐ โ = (5๐ ฬ + 2๐ ฬ โ 4๐ ฬ) + ๐ (3๐ ฬ + 2๐ ฬ โ 8๐ ฬ) Therefore, equation of line in vector form is ๐ โ = (5๐ ฬ + 2๐ ฬ โ 4๐ ฬ) + ๐ (3๐ ฬ + 2๐ ฬ โ 8๐ 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 (๐ฅ โ ๐ฅ1)/๐ = (๐ฆ โ ๐ฆ1)/๐ = (๐ง โ ๐ง1)/๐ Since line passes through (5, 2, โ4) ๐ฅ1 = 5, y1 = 2 , z1 = โ4 Also, line is parallel to 3๐ ฬ + 2๐ ฬ โ 8๐ ฬ , ๐ = 3, b = 2, c = โ 8 Equation of line in Cartesian form is (๐ฅ โ ๐ฅ1)/๐ = (๐ฆ โ ๐ฆ1)/๐ = (๐ง โ ๐ง1)/๐ (๐ฅ โ 5)/3 = (๐ฆ โ 2)/2 = (๐ง โ ( โ 4))/( โ 8) (๐ โ ๐)/๐ = (๐ โ ๐)/๐ = (๐ + ๐)/(โ๐)

Equation of line - given point and //vector

Chapter 11 Class 12 Three Dimensional Geometry

Concept wise

- Direction cosines and ratios
- Equation of line - given point and //vector
- Equation of line - given 2 points
- Angle between two lines - Vector
- Angle between two lines - Cartisian
- Angle between two lines - Direction ratios or cosines
- Shortest distance between two skew lines
- Shortest distance between two parallel lines
- Equation of plane - In Normal Form
- Equation of plane - Prependicular to Vector & Passing Through Point
- Equation of plane - Passing Through 3 Non Collinear Points
- Equation of plane - Intercept Form
- Equation of plane - Passing Through Intersection Of Planes
- Coplanarity of 2 lines
- Angle between two planes
- Distance of point from plane
- Angle between Line and Plane
- Equation of line under planes condition
- Point with Lines and Planes

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.