The vector equation of a line passing through the point $(1, -1, 0)$ and parallel to Y-axis is

$\vec{r} = \lambda\hat{j}$

The correct answer is Option (4) → $\vec{r} = \lambda\hat{j}$ ##

We know that vector eq. of a line is

$\vec{r} = \vec{a} + \lambda\vec{b}$

Given $\vec{a} = (1, -1, 0)$

Passing through the point line is parallel to y-axis:

$\vec{b} = (0, 1, 0)$

$\vec{r} = \hat{i} - \hat{j} + \lambda\hat{j}$