Analytical Plane Geometry

Find the angle \(\varphi \) between the lines \(3x - 7 = 0\) and \(x + y + 13 = 0\).

Find the angle \(\varphi \) between the lines \(p\) and \(q\) given by their parametric equations. \[ p\colon \begin{aligned}[t] x& = 1 + 2t, & \\y& = 3 - 3t;\ t\in \mathbb{R}, \\ \end{aligned}\qquad q\colon \begin{aligned}[t] x& = 2 - k, & \\y& = 3 + k;\ k\in \mathbb{R} \\ \end{aligned} \]

Given points \(A = [-1.4]\), \(B = [2,-2]\), \(C = [5,-1]\), find the angle \(\varphi \) between the lines \(AB\) and \(BC\).

Find the distance from the point \(M = [1;1]\) to the line \(p\). \[ \begin{aligned}p\colon x& = 3 + t, & \\y & = 1 + t;\ t\in \mathbb{R} \\ \end{aligned} \]