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LCDF

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- TL;DR Summary
- This post is an extension of the post: https://www.physicsforums.com/threads/probability-that-two-points-are-on-opposite-sides-of-a-line.994977/#post-6406442

In ##\mathbb{R}^2##, there are two lines passing through the origin that are perpendicular to each other. The orientation of one of the lines with respect to ##x##-axis is ##\psi \in [0, \pi]##, where ##\psi## is uniformly distributed in ##[0, \pi]##. Also, there are two vectors in ##\mathbb{R}^2## from the origin corresponding to points ##(x_1, y_1)## and ##(x_2, y_2)##. The angle between these vectors is ##\theta \in (0, 2\pi)## measured in anticlockwise direction, and has the probability density function ##f_{\theta}##. I want to calculate the probability that none of the perpendicular lines lies between the two vectors.

My attempt: I notice that this is equivalent to calculating the probability that the two points ##(x_1, y_1)## and ##(x_2, y_2)## lie on the same side of each of the perpendicular lines. In other words, a line does not lie between the two vectors iff ##(a_1x_1 + b_1y_1 +c_1)(a_1x_2 + b_1y_2 +c_2) > 0## AND ##(a_2x_1 + b_2y_1 +c_1)(a_2x_2 + b_2y_2 +c_2) > 0##, where ##a_1x + b_1y +c = 0## and ##a_2x + b_2y +c = 0## are the equations of the lines. But I am unable to obtain an expression for the probability.

For the case of only one line passing through the origin (ignore the second line), the probability that the line does not lie between two vectors is

##p = \frac{1}{\pi}\int_{0}^{\pi} (\pi -\rho) f_{\theta}(\rho)\mathrm{d}\rho + \frac{1}{\pi}\int_{\pi}^{2\pi} (\rho-\pi) f_{\theta}(\rho)\mathrm{d}\rho.##

My attempt: I notice that this is equivalent to calculating the probability that the two points ##(x_1, y_1)## and ##(x_2, y_2)## lie on the same side of each of the perpendicular lines. In other words, a line does not lie between the two vectors iff ##(a_1x_1 + b_1y_1 +c_1)(a_1x_2 + b_1y_2 +c_2) > 0## AND ##(a_2x_1 + b_2y_1 +c_1)(a_2x_2 + b_2y_2 +c_2) > 0##, where ##a_1x + b_1y +c = 0## and ##a_2x + b_2y +c = 0## are the equations of the lines. But I am unable to obtain an expression for the probability.

For the case of only one line passing through the origin (ignore the second line), the probability that the line does not lie between two vectors is

##p = \frac{1}{\pi}\int_{0}^{\pi} (\pi -\rho) f_{\theta}(\rho)\mathrm{d}\rho + \frac{1}{\pi}\int_{\pi}^{2\pi} (\rho-\pi) f_{\theta}(\rho)\mathrm{d}\rho.##