Exercises: Parallelism and perpendicularity

Two lines have gradients $m_1 = \frac{2}{3}$ and $m_2 = \frac{3}{2}$. A student concludes that, since they are formed by inverted digits, the lines are perpendicular. Explain the mistake in this "trick" and what the value of $m_2$ should be for the lines to be truly perpendicular.

Write the coordinates of a vector that is parallel to, and another that is perpendicular to, vector $\vec{u} = (-4, 7)$.

Given line $r$ with equation $y = -5x + 3$, state the gradient of any line parallel to $r$ and the gradient of any line perpendicular to $r$.

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

Determine the general equation of the line perpendicular to $3x + 4y - 2 = 0$ that passes through the origin.

Determine the relative position of lines $r: 2x - y + 4 = 0$ and $s: -4x + 2y - 8 = 0$.

Verify using vectors whether points $A(-1, 2)$, $B(2, 4)$ and $C(8, 8)$ are collinear.

Calculate the value of $k$ such that vectors $\vec{u} = (k, 3)$ and $\vec{v} = (4, -2)$ are perpendicular.

Obtain the equation of the line parallel to the y-axis passing through point $Q(-5, 2)$.

In an exam, Marcos must find the equation of a line perpendicular to $r: y = 2x - 3$ passing through the point $P(4, 1)$. His answer was as follows:

"Since it is perpendicular, the slope has the opposite sign, so $m = -2$. Using the point-slope equation: $y - 1 = -2(x - 4) \implies y = -2x + 9$".

Identify the conceptual mistake Marcos made when transforming the slope, explain geometrically why a $90^\circ$ rotation not only changes the sign but also inverts the slope, and write the correct explicit equation of the perpendicular line.

In the urban development plan for expanding a logistics platform in Zaragoza, a freight railway track follows a straight route modeled by the equation $r: 3x - y + 6 = 0$ (where the $x$ and $y$ coordinates are expressed in hundreds of meters). The engineering team must design a paved service road that is strictly parallel to the railway line to avoid level crossings and that provides access to the loading dock located at point $M(2, 5)$.

1. a) Find the slope of the railway track and justify what value the slope of the service road must have to ensure parallelism.
2. b) Find the explicit equation of the new service road.
3. c) If a maintenance vehicle starts from dock $M$ and travels east, increasing its $x$-coordinate by $4$ units ($400\text{ m}$), by how many units will its $y$-coordinate increase according to the calculated slope?

On a town plan, the main avenue follows the line $r: 2x - 5y + 10 = 0$. A pedestrian street parallel to the avenue is to be built, passing through the central square located at point $P(4, -1)$. Find the general equation of the new pedestrian street.

A laser beam travels along the line $x + 3y - 6 = 0$. A flat mirror is placed perpendicular to the beam at the exact point where it intersects the x-axis. Determine the explicit equation of the line representing the mirror.

Find the value of the parameter $m$ such that line $mx - 3y + 5 = 0$ is perpendicular to the line passing through points $A(2, -4)$ and $B(5, 1)$.

An agricultural plot of land is shaped like a parallelogram. Three of its vertices are located at points $A(1, 1)$, $B(5, 2)$ and $C(6, 6)$. Calculate the coordinates of the fourth vertex $D$, knowing that sides $AB$ and $CD$ are parallel, as are sides $AD$ and $BC$.

Calculate the general equation of the perpendicular bisector of the line segment whose endpoints are points $A(-3, 5)$ and $B(1, -1)$.

Two ships follow straight-line paths. Ship A moves along line $r: 4x + ky - 7 = 0$ and Ship B along line $s: x - 2y + 5 = 0$. Determine the value of $k$ so that their paths are parallel, and check whether they could collide in that case (that is, whether the lines are coincident).

In the triangle with vertices $A(-2, 1)$, $B(4, 3)$ and $C(2, -3)$, find the general equation of the altitude from vertex $A$.

Determine the components of a vector $\vec{u}$ that is perpendicular to vector $\vec{v} = (-6, 8)$ and has a magnitude of exactly $15$ units.

A math teacher wants to design a multiple-choice exam question where the correct answer is: "The equation of the perpendicular bisector of segment $AB$ is $s: y = -0,5x + 3$".

1. a) If the teacher decides to fix one of the endpoints at point $A(1, 0)$, reconstruct step by step the coordinates that the other endpoint $B(x_B, y_B)$ must have so that line $s$ is strictly its perpendicular bisector (remember that the perpendicular bisector is perpendicular to the segment and passes through its midpoint).
2. b) Check your result by verifying that the distances from $A$ and from $B$ to the $y$-intercept of the perpendicular bisector, $P(0, 3)$, are exactly equal.

On an irrigated farm in Albacete, an underground main pipe connects two wells located at coordinates $P_1(1, 2)$ and $P_2(7, 5)$ (measured in hectometers, $\text{hm}$). There are plans to install a straight secondary pipe perpendicular to the main one that supplies water to an auxiliary tank located at $D(3, 8)$.

1. a) Estimate before calculating: Observing that the main pipe ascends gently (it advances $6\text{ hm}$ horizontally to rise only $3\text{ hm}$ vertically), should the slope of the perpendicular pipe be positive or negative? Will its absolute value be greater than $1$ (very steep) or less than $1$ (not very steep)? Reasonably estimate its sign and steepness.
2. b) Calculate and check: Determine the exact slope of the main pipe and that of the secondary pipe. Check whether it matches your previous estimate.
3. c) Write the general equation of the secondary pipe.
4. d) If this pipe is extended until it crosses the southern boundary of the property (the line $y = 0$), at what $x$-coordinate will the crossing occur? Is it within the boundaries of the land if the property spans from $x = 0$ to $x = 10\text{ hm}$?

Two guy cables of a suspension bridge are modelled in the Cartesian plane. The first cable follows line $r: ax - 3y - 4 = 0$ and the second passes through points $P(1, 3)$ and $Q(a, 5)$. Determine the possible values of the parameter $a$ for the cables to be parallel, and the value of $a$ for them to be perpendicular.

Find the coordinates of the orthocentre (intersection point of the altitudes) of the triangle whose vertices are $A(0, 4)$, $B(-4, 0)$ and $C(2, -2)$.

In a video game, a character fires a projectile from point $P(1, 6)$ in a direction perpendicular to an energy shield. The shield moves along the line joining points $M(3, 2)$ and $N(7, -1)$. Calculate the exact coordinates of the impact point of the projectile on the line of the shield.

Given the line $r: 2x - y + 3 = 0$ and the point $A(4, 1)$, find the coordinates of point $A'$, which is the reflection of $A$ in the line $r$.

A ray of light starts from point $A(1, 5)$, strikes a flat mirror along the x-axis at point $P(x, 0)$ and is reflected, passing through point $B(7, 2)$. Given that the angle of incidence equals the angle of reflection, calculate the exact coordinates of the impact point $P$.

Determine the values of the parameters $A$ and $B$ so that the lines $r: Ax + By - 5 = 0$ and $s: 3x - 4y + 1 = 0$ are perpendicular and intersect at a point whose y-coordinate is $y = 1$.

A square $ABCD$ has side $AB$ on the line $r: x + 3y - 5 = 0$. Given that vertex $C$ has coordinates $C(4, 7)$, find the equation of the line containing side $BC$ and calculate the area of the square.

Prove analytically that the quadrilateral with vertices $A(-1, -2)$, $B(4, -1)$, $C(5, 4)$ and $D(0, 3)$ is a rhombus. To do this, check that opposite sides are parallel and that its diagonals are perpendicular to each other.

Consider the following two lines expressed in general form:

$r: 3x - 4y + 5 = 0$

$s: 4x + 3y - 12 = 0$

Two students discuss which is the most efficient method to prove whether they are perpendicular:

• Laura's method (slopes): Solve for the variable $y$ in both equations to obtain their explicit slopes ($m_r$ and $m_s$) and check if $m_r \cdot m_s = -1$.

• Carmen's method (direction vectors): Directly obtain a direction vector of each line from the general coefficients of the equation ($Ax + By + C = 0 \implies \vec{v} = (-B, A)$) and calculate their dot product $\vec{v_r} \cdot \vec{v_s}$ to check if it equals zero.

1. a) Solve the problem by applying both methods step by step.
2. b) Compare the two procedures: which one is faster and reduces the risk of errors with signs or fractions when the lines are given in general form? Justify your answer.
3. c) If you are now given the lines in symmetric (continuous) form $\dfrac{x - 2}{5} = \dfrac{y + 1}{-3}$ and $\dfrac{x + 4}{3} = \dfrac{y}{5}$, which of the two methods would you choose to use directly, and why?

The Maritime Rescue coordination center in Finisterre (Galicia) records two trajectories on its radar screens during a stormy day (coordinates in nautical miles):

• A fishing boat follows the straight route $r_1: 3x + 4y - 12 = 0$.

• A cargo ship sails along a safety route $r_2$ strictly parallel to $r_1$, passing through an oceanographic buoy located at $B(10, 5)$.

• A fast rescue boat departs from the coastal base at the origin $O(0, 0)$ to intercept the cargo ship's trajectory $r_2$ along the shortest possible straight-line distance (that is, a trajectory perpendicular to $r_2$).

1. a) Model the general equation of the cargo ship's route $r_2$ and justify analytically why it will not collide with the fishing boat's route.
2. b) Determine the equation of the straight trajectory described by the rescue boat from base $O(0, 0)$.
3. c) Analytically calculate the exact coordinates of the expected meeting point on the radar between the rescue boat and the cargo ship's route.
4. d) Find the total distance (in nautical miles) that the rescue boat will travel from the coast to that point and interpret whether the chosen trajectory optimizes fuel consumption and response time.

Topic challenge. How would you design a relief drone logistics network for your municipality?

1. Data collection. Go to Google Maps or the Iberpix (IGN) viewer and set your school as the origin $(0,0)$. Locate two points of interest in your neighbourhood (for example, health centre $A$ and sports centre $B$) and obtain their coordinates $(x,y)$ in kilometres using the graphic scale.

2. Main axis and intermediate base. Calculate the components of vector $\vec{AB}$ and the real Euclidean distance between both points. Determine the midpoint $M$, where a safe recharging station will be located.

3. Route plotting. Find the slope $m$ and the equation of the line connecting $A$ and $B$. Design a secondary corridor parallel to it that passes through the school $(0,0)$ and a perpendicular emergency route that intersects the main one at $M$.

4. Technical report. Plot the entire system in GeoGebra (points, vectors, and lines) and write a short half-page report justifying the geometric feasibility of the network.

Assessment criteria:

• Precision in calculating coordinates, vectors, midpoint, and distances.

• Accuracy in finding slopes and applying the conditions for parallelism and perpendicularity.

• Quality of the design in GeoGebra and thoroughness in the report's conclusions.