The vector equation of a line passing through a point with position vector \(\vec{a}\) and parallel to a vector \(\vec{b}\) is given by: \(\vec{r} = \vec{a} + \lambda\vec{b}\), where \(\lambda\) is a scalar.

Here, the line passes through the point (1, -1, 0), so the position vector \(\vec{a}\) is given by: \(\vec{a} = \hat{i} - \hat{j} + 0\hat{k} = \hat{i} - \hat{j}\)

The line is parallel to the Y-axis, so the direction vector \(\vec{b}\) is given by: \(\vec{b} = \hat{j}\)

Therefore, the vector equation of the line is: \(\vec{r} = (\hat{i} - \hat{j}) + \lambda\hat{j}\)