Question

The Cartesian equation of a line is $fraction numerator straight x plus 3 over denominator 2 end fraction space equals space fraction numerator straight y minus 5 over denominator 4 end fraction space equals fraction numerator straight z plus 6 over denominator 2 end fraction.$ Find the vector equation for the line.

Solution

The Cartesian equation of line is

fraction numerator straight x plus 3 over denominator 2 end fraction space equals space fraction numerator straight y minus 5 over denominator 4 end fraction space equals space fraction numerator straight z plus 6 over denominator 2 end fraction

fraction numerator straight x minus left parenthesis negative 3 right parenthesis over denominator 2 end fraction space equals fraction numerator straight y minus 5 over denominator 4 end fraction space equals space fraction numerator straight z minus left parenthesis negative 6 right parenthesis over denominator 2 end fraction

∴ line passes through (– 3, 5, – 6) and has direction ratios 2, 4, 2.

therefore space space space straight a with rightwards arrow on top space equals space minus 3 space straight i with hat on top space plus space 5 space straight j with hat on top space minus space 6 space straight k with hat on top space space and space space straight m with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 2 space straight k with hat on top

Let

straight r with rightwards arrow on top

be the position vector of any point on the given line.
∴ the vector equation of line is

straight r with rightwards arrow on top space equals space straight a with rightwards arrow on top space plus space straight lambda straight m with rightwards arrow on top

straight r with rightwards arrow on top space equals space left parenthesis negative 3 space straight i with overparenthesis on top space plus space 5 space straight j with overparenthesis on top space minus space 6 space straight k with overparenthesis on top right parenthesis plus space straight lambda space left parenthesis 2 straight i with hat on top space plus space 4 straight j with hat on top space plus space 2 straight k with hat on top right parenthesis

For Daily Free Study Material Join wiredfaculty