Find the Cartesian equation of a line passing through the points `A(2,\ -1,\ 3)a n d\ B(4,\ 2,\ 1)dot` Also reduce it to vector form.

Explanation

`A(2,-1,3), B(4,2,1)`


`frac{x-x_{1}}{x_{2}-x_{1}}=frac{y-y_{1}}{y_{2}-y_{1}}=frac{z-z_{1}}{z_{2}- z_{1}}`


`frac{x-2}{4-2}=frac{y+1}{2+1}=frac{z-3}{1-3}`


`frac{x-2}{2}=frac{y+1}{3}=frac{z-3}{-2} rightarrow `Cartesian form


`frac{x-2}{2}=frac{y+1}{3}=frac{z-3}{-2}=lambda`


`x=2 lambda+2 quad y=3 lambda-1, z=-2 lambda+3`


`vec{r}=2 hat{i}+y hat{j}+2 hat{k}`


`=(2 lambda+2) hat{i}+(3 lambda-1) hat{j}+(-2 lambda+3) hat{k} .`


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