Let the position vector of the point be $\vec{a} = 2\hat{i} - \hat{j} + 4\hat{k}$ and the direction vector of the line be $\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$.

The vector equation of the line passing through a point with position vector $\vec{a}$ and parallel to vector $\vec{b}$ is given by $\vec{r} = \vec{a} + \lambda \vec{b}$. Substituting the given vectors, we get:

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

For the Cartesian form, the coordinates of the given point are $(2, -1, 4)$ and the direction ratios are $(1, 2, -1)$. Using the formula $\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$, we get:

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