Angle between two lines. Parallel and perpendicular lines

Calculate the acute angle between the lines $r: 2x - y + 4 = 0$ and $s: 3x + y - 1 = 0$.

Calculate the angle formed by the lines given by their explicit equations $y = 2x + 1$ and $y = -3x + 4$.

Determine the angle that the line $\sqrt{3}x - y + 2 = 0$ forms with the positive direction of the x-axis.

Determine the general equation of the line parallel to $y = 3x - 5$ passing through the point $P(2, -1)$.

Given the lines $r: kx - 2y + 5 = 0$ and $s: 3x + 4y - 1 = 0$, find the value of the parameter $k$ so that both lines are parallel.

Find the equation of the line perpendicular to $x + 4y - 7 = 0$ passing through the origin of coordinates. Express the result in explicit form.

Find the value of $m$ so that the lines $r: 5x + 2y = 0$ and $s: mx - 5y + 3 = 0$ are perpendicular.

State which of the following lines are parallel and which are perpendicular to each other:

• $r_1: 2x - 3y + 1 = 0$

• $r_2: 6x + 4y - 5 = 0$

• $r_3: -4x + 6y + 7 = 0$

Calculate the values of the parameter $k$ so that the line $kx + y - 2 = 0$ forms an angle of $45^\circ$ with the line with equation $x - 2y + 1 = 0$.

Two aeroplanes follow straight-line paths on a radar screen given by the equations $r: x - 3y + 2 = 0$ and $s: 2x + ky - 5 = 0$. Determine the value (or values) of $k$ so that their routes cross, forming an exact angle of $60^\circ$.

Two robotic arms in a factory move along straight rails. The first follows the line $r: 3x + 4y - 12 = 0$ and the second the line $s: ax + 8y - 5 = 0$.

1. Determine $a$ so that the rails are parallel.

2. Find the equation of a third transverse rail that is perpendicular to both and crosses rail $r$ exactly at its y-intercept.

Consider the family of lines $r_k: kx + (k - 1)y - 3 = 0$, where $k \in \mathbb{R}$, and the fixed line $s: 2x - y + 1 = 0$.

1. a) Determine the value of $k$ for which $r_k$ is perpendicular to $s$, verifying the condition both with the product of slopes and using the dot product of direction vectors.
2. b) Find the value of $k$ for which $r_k$ and $s$ are parallel.
3. c) It is desired that the acute angle between $r_k$ and $s$ be exactly $\alpha = \frac{\pi}{4}\text{ rad}$ ($45^\circ$). Set up the corresponding equation (using slopes or vectors), solve it, and determine how many values of $k$ satisfy this condition.
4. d) Express the equation of the family in the form $A(x,y) + k\times B(x,y) = 0$ to prove that all lines $r_k$ pass through the same fixed point $Q$ (pencil of concurrent lines). Using this geometric property, justify why it was expected to obtain exactly two values of $k$ in section c), one in section a), and one in section b).

In the design of a park, a new avenue must be perpendicular to the main street defined by the equation $3x - 4y + 10 = 0$. Furthermore, this avenue must pass through the point of intersection of the side streets $x + y - 2 = 0$ and $2x - y - 4 = 0$. Obtain the general equation of the new avenue.

Given a triangle with vertices $A(1, 2)$, $B(5, 3)$ and $C(4, 6)$, find the general equation of the altitude corresponding to vertex $A$ and calculate the angle this altitude forms with the side $AB$.

In the plan for a metallic structure, a main support beam rests on the line passing through nodes $P(-2, 4)$ and $Q(4, 1)$. A second beam needs to be welded perpendicular to the first exactly at its midpoint. Find the explicit equation of the line defining this second beam.