Exercises: Angle between two lines. Parallelism and perpendicularity

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

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

Find the equation of the line perpendicular to $x + 4y - 7 = 0$ that passes through the origin. Express the result in slope-intercept form.

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

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

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

• $r_1: 2x - 3y + 1 = 0$

• $r_2: 6x + 4y - 5 = 0$

• $r_3: -4x + 6y + 7 = 0$

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

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 secondary streets $x + y - 2 = 0$ and $2x - y - 4 = 0$. Find the general equation of the new avenue.

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

A laser beam travels along the line $y = -2x + 5$ and reflects off a plane mirror located on the x-axis. Calculate the angle of incidence (angle between the beam and the normal to the mirror) and determine the equation of the line containing the reflected beam.

Two airplanes follow rectilinear trajectories 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 $60^\circ$.

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

On the plan of a metal structure, a main support beam rests on the line passing through the 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 slope-intercept equation of the line that defines this second beam.

Find the coordinates of the orthogonal projection of the origin $(0,0)$ onto the line of equation $2x - 5y + 10 = 0$.

Two robotic arms in a factory move on rectilinear 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 point of intersection with the y-axis.

Find the equations of the angle bisectors of the angles formed by the lines $r: 3x - 4y + 5 = 0$ and $s: 12x + 5y - 1 = 0$. Prove analytically, using their direction vectors or their slopes, that both angle bisectors are perpendicular to each other.

The diagonals of a rhombus are contained in the lines $r: x + 2y - 4 = 0$ and $s: 2x - y - 3 = 0$. Knowing that one of its vertices is $A(0, 2)$ and that the length of its side is $\sqrt{20}$, find the coordinates of the other three vertices.

A vertex of an equilateral triangle is the origin $(0,0)$ and the opposite side is contained in the line $x + y - 4 = 0$. Determine the general equations of the lines containing the other two sides of the triangle knowing that they must form an angle of $60^\circ$ with the given side.

Given the family of lines depending on the parameter $k \in \mathbb{R}$: $(2 + k)x + (1 - 2k)y + 3k - 4 = 0$. Prove that all the lines in this family pass through the same fixed point and calculate its coordinates. Then, find the value of $k$ to obtain the line in the family that is perpendicular to $3x - y + 2 = 0$.

A solar panel is aligned with the line $r: \sqrt{3}x - y + 10 = 0$. The sun's rays fall parallel to the line $s: 2x + 2y - 5 = 0$. Calculate the angle of incidence of the rays on the panel. If the panel must be reoriented by pivoting on its point of intersection with the $OY$-axis so that the rays fall completely perpendicular to its surface, what will be the new general equation of the line that defines the panel?

Two opposite vertices of a square are $A(-1, 3)$ and $C(5, 1)$. Find the equations of the lines containing the four sides of the square, first calculating its center and the directions of its diagonals.

Determine the general equation of the line symmetric to $r: 2x - 3y + 6 = 0$ with respect to the line $s: x + y - 1 = 0$.

Prove that if two lines $r: ax + by + c = 0$ and $s: a'x + b'y + c' = 0$ intersect forming an angle $\alpha$, then $\tan \alpha = \left| \frac{ab' - a'b}{aa' + bb'} \right|$ is satisfied. Use this result to find the general equations of the lines that pass through $P(1,1)$ and form an angle $\alpha$ such that $\tan \alpha = \frac{1}{2}$ with the line $2x - y + 3 = 0$.