Find the equation of the line in vector and in Cartesian form that passes through the point with position vector

It is given that the line passes through the point with position vector

$\vec{a}=2 \hat{i}-\hat{j}+4 \hat{k}$                    ...(1)

$\vec{b}=\hat{i}+2 \hat{j}-\hat{k}$                    $\ldots(2)$

It is known that a line through a point with position vector $\vec{a}$ and parallel to $\vec{b}$ is given by the equation, $\vec{r}=\vec{a}+\lambda \vec{b}$

$\Rightarrow \vec{r}=2 \hat{i}-\hat{j}+4 \hat{k}+\lambda(\hat{i}+2 \hat{j}-\hat{k})$

This is the required equation of the line in vector form.

$\vec{r}=x \hat{i}-y \hat{j}+z \hat{k}$

$\Rightarrow x \hat{i}-y \hat{j}+z \hat{k}=(\lambda+2) \hat{i}+(2 \lambda-1) \hat{j}+(-\lambda+4) \hat{k}$

Eliminating λ, we obtain the Cartesian form equation as

$\frac{x-2}{1}=\frac{y+1}{2}=\frac{z-4}{-1}$

This is the required equation of the given line in Cartesian form.