50.
Find the vector and the Cartesian equations for the line through the points A (3, 4, –7) and B (5, 1, 6).
The line passes through the points A (3, 4, –7) and B (5, 1,6).
![therefore space space space space space space space straight a with rightwards arrow on top space equals space 3 straight i with hat on top space plus space 4 space straight j with hat on top space minus space 7 space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space 5 space straight i with hat on top space plus space straight j with hat on top space plus 6 space straight k with hat on top](/application/zrc/images/qvar/MAEN12064440.png)
where
are position vectors of A, B respectively.
![straight b with rightwards arrow on top minus straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space 13 space straight k with hat on top](/application/zrc/images/qvar/MAEN12064440-3.png)
Let
be the position vector of any point on the given line.
∴ the vector equation of the line is
![straight r with rightwards arrow on top space equals space straight a with rightwards arrow on top space plus space straight lambda left parenthesis straight b with rightwards arrow on top minus stack straight a right parenthesis with rightwards arrow on top](/application/zrc/images/qvar/MAEN12064440-5.png)
or ![straight r with rightwards arrow on top space equals space 3 straight i with hat on top space plus space 4 straight j with hat on top space minus space 7 straight k with hat on top space plus space straight lambda left parenthesis 2 straight i with hat on top space minus space 3 straight j with hat on top space plus space 13 straight k with hat on top right parenthesis](/application/zrc/images/qvar/MAEN12064440-6.png)
Taking ![straight r with rightwards arrow on top space equals space straight x space straight i with hat on top space plus space straight y space straight j with hat on top space plus space straight z space straight k with hat on top comma space space the space equation space of space line space is](/application/zrc/images/qvar/MAEN12064440-7.png)
![straight x space straight i with hat on top space plus straight y space straight j with hat on top space space plus space straight z space straight k with hat on top space equals space left parenthesis 3 space plus space 2 space straight lambda right parenthesis space straight i with hat on top space plus space left parenthesis 4 minus 3 space straight lambda right parenthesis space straight j with hat on top space space plus space left parenthesis negative 7 plus 13 space straight lambda right parenthesis space straight k with hat on top](/application/zrc/images/qvar/MAEN12064440-8.png)
Comparing the coefficients of
we get,
![straight x space equals space 3 plus 2 space straight lambda comma space space space straight y space equals space 4 minus 3 straight lambda comma space space space straight z space equals space minus 7 space plus space 13 space straight lambda](/application/zrc/images/qvar/MAEN12064440-10.png)
![rightwards double arrow space space space space space space space space fraction numerator straight x minus 3 over denominator 2 end fraction space equals space fraction numerator straight y minus 4 over denominator negative 3 end fraction space equals space fraction numerator straight z plus 7 over denominator 13 end fraction space left parenthesis space equals space straight lambda right parenthesis](/application/zrc/images/qvar/MAEN12064440-11.png)
which is the Cartesian equation of line.
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