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Three Dimensional Geometry

Question
CBSEENMA12033251

The cartesian equations of a line are fraction numerator straight x minus 5 over denominator 3 end fraction space equals space fraction numerator straight y plus 4 over denominator 7 end fraction space equals fraction numerator straight z minus 6 over denominator 2 end fraction. Find a vector equation for the line.

Solution
The cartesian equations of line are
      fraction numerator straight x minus 5 over denominator 3 end fraction space equals fraction numerator straight y plus 4 over denominator 7 end fraction space equals fraction numerator straight z minus 6 over denominator 2 end fraction
∴   line passes through (5, - 4, 6) and has direction-ratios 3, 7, 2
space space space therefore space space space space straight a with rightwards arrow on top space equals space 5 straight i with hat on top space minus space 4 straight j with hat on top space plus space 6 straight k with hat on top space space and space space straight m with rightwards arrow on top space equals space 3 straight i with hat on top plus 7 straight j with hat on top space plus space 2 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 plus space straight lambda space straight m with rightwards arrow on top
or       straight r with rightwards arrow on top space equals space left parenthesis 5 space straight i with bar on top space minus space 4 stack space straight j with hat on top space plus space 6 straight k with hat on top right parenthesis space plus space straight lambda left parenthesis 3 space 1 with overparenthesis on top space plus space 7 space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis
where λ is a parameter.

Some More Questions From Three Dimensional Geometry Chapter

Find the direction cosines of x, y and z-axis.