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.

Seán has to calculate the measure of the angle formed by the lines $r: 2x - y + 3 = 0$ and $s: 3x + y - 5 = 0$. To do so, he solves for their slopes, obtaining $m_r = 2$ and $m_s = -3$. Next, he applies the angle formula:

Seán deduces that $\alpha = \arctan(-1) = -45^\circ$ and reasons: "Since a geometric angle between lines cannot be negative, I add $180^\circ$ to it and the solution is $135^\circ$".

1. a) Point out the two conceptual errors made by Seán (one in the formula applied itself and another when interpreting which angle defines the relative position of two intersecting lines).
2. b) Explain why in Euclidean geometry the angle between two lines is always taken to be non-obtuse, that is, bounded in the interval $[0^\circ, 90^\circ]$ (or $[0, \pi/2]\text{ rad}$).
3. c) Correct the procedure and calculate the correct measure of the acute angle, expressing it both in sexagesimal degrees and in radians (in terms of $\pi$).

Consider the lines $r: y = \frac{\sqrt{3}}{3}x + 1$ and $s: y = \sqrt{3}x - 4$.

1. a) Without using the formulas for calculating the angle between lines yet, relate the slope of each line to its angle of inclination with respect to the positive x-semiaxis ($m = \tan \theta$). From this relationship, make a reasoned estimation of the exact value of the acute angle $\alpha$ that they form with each other (both in sexagesimal degrees and in radians).
2. b) Analytically verify your estimation by calculating the angle using the dot product of their direction vectors with the expression $\cos \alpha = \frac{|\vec{u}\cdot\vec{v}|}{|\vec{u}||\vec{v}|}$.
3. c) Check the result again by applying the tangent formula from the slopes: $\tan \alpha = \left|\frac{m_s - m_r}{1 + m_s \times m_r}\right|$. Why do all three methods agree and what is the usefulness of estimating before calculating?

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.

Consider the lines in the plane $r: x - \sqrt{3}y + 6 = 0$ and the vertical line $s: x = -3$.

1. a) Try to calculate the acute angle between $r$ and $s$ using the slope formula $\tan \alpha = \left|\frac{m_r - m_s}{1 + m_r \times m_s}\right|$. Explain what mathematical limitation or indeterminate form arises when assigning a numerical value to $m_s$ and why this formula is not directly workable.
2. b) Set up the calculation using the direction vectors $\vec{u}_r$ and $\vec{u}_s$ along with the cosine formula. Justify why the vector approach is more general and advantageous in this case.
3. c) Obtain the value of the acute angle $\alpha$ using the vector method and express it both in sexagesimal degrees and in radians (as an irreducible fraction of $\pi$).

In a railway infrastructure project by Irish Rail in the surroundings of Dublin, the main track runs along the line $r: 3x - 4y + 12 = 0$ (on a coordinate plane in kilometres). A new branch link to a logistics terminal is planned along the line $s: x - 7y - 5 = 0$.

The regulations of the Commission for Railway Regulation (CRR) state that if the acute angle of divergence between two alignments is greater than $0.26\text{ rad}$ (approximately $15^\circ$), a direct turnout at conventional speed is not acceptable and a transition curve with prior deceleration must be designed.

1. a) Model the situation by calculating the acute angle $\alpha$ formed by both tracks. Determine the exact value of $\tan \alpha$ and the measure of $\alpha$ in sexagesimal degrees and in radians (to three decimal places).
2. b) Rule, justifying the conclusion with the previous calculation, whether the infrastructure mandatorily requires such a deceleration curve.
3. c) A secondary escape track $t$ is planned, which starts at the point where $r$ intersects the x-axis and is strictly perpendicular to the main track $r$. Obtain the general equation of $t$, checking the perpendicularity using the relationship between their slopes.

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 centre 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 $2x - y + 3 = 0$.

A communications antenna is located at the point $P(2, 1)$. A straight underground fibre optic line, represented by $s$, needs to be installed so that it forms an acute angle of $\frac{\pi}{4}\text{ rad}$ ($45^\circ$) with an existing water pipeline that follows the line $r: 2x - y + 5 = 0$.

1. a) Set up the equation that the slope $m$ of line $s$ must satisfy using the formula $\tan \alpha = \left|\frac{m - m_r}{1 + m \times m_r}\right|$. When algebraically solving the absolute value equation, justify why two possible slopes arise and calculate them.
2. b) Obtain the general equations of the two solution lines, $s_1$ and $s_2$, that pass through the point $P$.
3. c) Prove that $s_1$ and $s_2$ are perpendicular to each other by checking both the product of their slopes and the dot product of their direction vectors.
4. d) Geometrically interpret the result from section c): reason, without resorting to algebraic calculation, why two lines forming the same $45^\circ$ angle on either side of a given reference line must necessarily be perpendicular to each other.

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).