Step-by-Step Solution

The direction ratios of the line \(\frac{x}{1}=\frac{y}{-1}=\frac{z}{0}\) are 1, -1, 0.

The direction ratios of the Y-axis are 0, 1, 0.

Let \(\theta\) be the angle between the line and the Y-axis. Then, \[\cos \theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}}\] where \(a_1, b_1, c_1\) are the direction ratios of the line and \(a_2, b_2, c_2\) are the direction ratios of the Y-axis.

Substituting the values, we get \[\cos \theta = \frac{(1)(0) + (-1)(1) + (0)(0)}{\sqrt{1^2 + (-1)^2 + 0^2} \sqrt{0^2 + 1^2 + 0^2}} = \frac{-1}{\sqrt{2} \cdot 1} = -\frac{1}{\sqrt{2}}\]

Therefore, \(\theta = \cos^{-1}\left(-\frac{1}{\sqrt{2}}\right)\). Since we are looking for the angle with the positive direction of the Y-axis, we want the angle between 0 and \(\pi\).

The angle whose cosine is \(-\frac{1}{\sqrt{2}}\) in the range \([0, \pi]\) is \(\frac{3\pi}{4}\).