Exercises: Angle between two lines. Parallelism and perpendicularity

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

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

Find the equation of the line perpendicular to $x + 4y - 7 = 0$ passing through the origin of coordinates. Express the result in explicit 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 explicit equations $y = 2x + 1$ and $y = -3x + 4$.

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

• $r_1: 2x - 3y + 1 = 0$

• $r_2: 6x + 4y - 5 = 0$

• $r_3: -4x + 6y + 7 = 0$

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 side streets $x + y - 2 = 0$ and $2x - y - 4 = 0$. Obtain 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 with equation $x - 2y + 1 = 0$.

A laser beam travels along the line $y = -2x + 5$ and reflects off a flat mirror situated 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 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 $60^\circ$.

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.

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

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.

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 gradients, that both bisectors are perpendicular to each other.

The diagonals of a rhombus lie on 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 its side length is $\sqrt{20}$, find the coordinates of the other three vertices.

One vertex of an equilateral triangle is the origin $(0,0)$ and the opposite side lies on 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 $k \in \mathbb{R}$: $(2 + k)x + (1 - 2k)y + 3k - 4 = 0$. Show 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 strike 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 about its intersection point with the $OY$-axis so that the rays hit completely perpendicular to its surface, what will be the new general equation of the line defining 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 at an angle $\alpha$, then $\tan \alpha = \left| \frac{ab' - a'b}{aa' + bb'} \right|$ holds. Use this result to find the general equations of the lines passing through $P(1,1)$ and forming an angle $\alpha$ such that $\tan \alpha = \frac{1}{2}$ with the line $2x - y + 3 = 0$.