Find the vector and the cartesian equations of a line that passes through the point $\displaystyle \mathrm{A}(1,2,-1)$ and parallel to the line $\displaystyle 5 x-25=14-7 \mathrm{y}=35 \mathrm{z}$.

The given line is $$\frac{x-5}{\dfrac{1}{5}}=\frac{\mathrm{y}-2}{-\dfrac{1}{7}}=\frac{\mathrm{z}}{\dfrac{1}{35}}, \text { or } \frac{x-5}{7}=\frac{\mathrm{y}-2}{-5}=\frac{\mathrm{z}}{1} So, the required vector equation of the line passing through $\displaystyle (1,2,-1)$ is \vec{r}=(\hat{\imath}+2 \hat{\jmath}-\hat{k})+\lambda(7 \hat{\imath}-5 \hat{\jmath}+\hat{k}) $$