Exercises: Coordinates

Calculate the components of the vector $\vec{AB}$, knowing that its initial point is $A(-3, 5)$ and its terminal point is $B(4, -2)$.

Determine the magnitude of the vector $\vec{v} = (6, -8)$. Simplify the result if possible.

Find the exact distance between the points $P(1, 4)$ and $Q(-5, -4)$ in the Cartesian plane.

Calculate the coordinates of the midpoint of the line segment whose endpoints are $M(7, -1)$ and $N(-3, 9)$.

Given the vectors $\vec{u} = (2, -5)$ and $\vec{v} = (-1, 3)$, calculate the coordinates of the vector resulting from the operation $3\vec{u} - 2\vec{v}$.

Find the coordinates of the point $A'$, knowing that it is the reflection of $A(-2, 6)$ in the centre of symmetry $P(1, 1)$.

Check analytically whether the points $A(1, 2)$, $B(3, 5)$ and $C(7, 11)$ are collinear by calculating the components of the vectors $\vec{AB}$ and $\vec{BC}$.

Given the vector $\vec{w} = (k, 3)$, calculate the possible values of the parameter $k$ such that the magnitude of this vector is exactly $5$.

To determine the displacement vector between two hiking beacons located at $A(-2, 5)$ and $B(4, 1)$, Mario performs the following calculation in his notebook: $\vec{AB} = (-2 - 4,\ 5 - 1) = (-6, 4)$. From this result, Mario states that to go from beacon $A$ to beacon $B$ one must travel $6\text{ km}$ west (left) and $4\text{ km}$ north (up).

1. a) Identify and explain in detail the conceptual error Mario made when applying the formula for the components of a vector.
2. b) Without applying numerical formulas yet, observe the coordinates of the origin at $A$ and the destination at $B$, and explain why Mario's qualitative conclusion about the direction and sense of the route makes no sense.
3. c) Write down the correct procedure and calculate the true components of the vector $\vec{AB}$.

The State Meteorological Agency (AEMET) has two automatic monitoring stations in an agricultural region of Castilla y León, represented on a coordinate map in kilometres: the Saldaña station at point $A(2, 3)$ and the Carrión station at point $B(8, 11)$.

1. a) AEMET plans to install a new digital rain gauge at the exact midpoint of the straight path connecting both facilities. Reasonably determine the coordinates where the rain gauge should be located.
2. b) A maintenance technician claims that the straight-line distance between both weather stations exceeds $12\text{ km}$. Mathematically verify whether this statement is true or false by calculating the exact magnitude of the vector $\vec{AB}$.

A forest surveillance drone takes off from point $A(-2, 1)$ on a grid map with units in kilometres. It flies in a straight line to point $B(4, 9)$ to inspect a suspicious clearing, and from there moves directly to point $C(10, 1)$ to examine a shelter. Calculate the total distance covered by the drone during this journey.

In a pirate exploration video game, the treasure chest is hidden at a point $T$ lying exactly on the straight line joining Skull Rock $R(-4, -1)$ and the Great Tree $A(6, 4)$. The clues indicate that the distance from the Rock to the treasure is twice the distance from the treasure to the Tree. Determine the exact coordinates of point $T$.

The geometric centre of a circular ice rink is located at coordinates $C(2, -3)$. If one end of the rink, where the entrance gate is located, is at point $A(-1, 1)$, find the coordinates of the diametrically opposite end $B$ and calculate the total diameter of the rink.

An architect is designing a suspension bridge in which three of the main cable anchors must be perfectly collinear to ensure correct tension. On the construction plan, their coordinates are $P(-3, 2)$, $Q(1, 4)$ and $R(a, 7)$. Calculate the value of parameter $a$ so that the structure is stable.

Two forces in the plane represented by vectors $\vec{F_1} = (4, 3)$ N and $\vec{F_2} = (-1, 5)$ N act on a point object subjected to a physics experiment. Calculate the magnitude of the resultant force $\vec{F_R}$ acting on the object.

The vertices of a triangular plot allocated for a new park are $A(-3, -2)$, $B(5, 0)$ and $C(1, 8)$. The local council wants to place a large central streetlight exactly at the centroid of the plot to optimise illumination. Calculate the coordinates where the streetlight should be installed.

Prove analytically, by calculating the exact length of each of its sides, that the triangle formed by the points $A(2, 1)$, $B(5, 5)$ and $C(-2, 4)$ is an isosceles triangle.

Determine the components of a vector $\vec{v}$ that is parallel to the vector $\vec{u} = (-3, 4)$, but has a magnitude of $15$ units and points in the opposite direction to $\vec{u}$.

During a time trial stage of the Vuelta Ciclista a España, the organisation's GPS device records a cyclist passing an intermediate timing point $A(1, 2)$ and, minutes later, passing the control point $B(7, 10)$, on a tracking grid where each unit represents $1\text{ km}$.

1. a) Reasonably estimate the straight-line distance between both points before calculating: knowing that the horizontal displacement is $6\text{ km}$ and the vertical displacement is $8\text{ km}$, justify why the straight-line distance must necessarily be greater than $8\text{ km}$ and less than $14\text{ km}$.
2. b) Calculate the exact magnitude of the displacement vector $\vec{AB}$ and check whether the value obtained matches your previous estimate.
3. c) If the bicycle's odometer indicates that the cyclist has travelled a distance on the asphalt of $9,4\text{ km}$ between $A$ and $B$, is this reading physically possible in reality? Justify your answer by comparing the geometric straight-line distance with the actual distance travelled along the road.

In the distribution network of a courier company, the regional logistics centre $M(3, 4)$ is known to be located at the exact midpoint of the straight route connecting the shipping warehouse $A(-1, 9)$ to a delivery point $B$.

1. a) Set up the corresponding equations from the definition of a midpoint and deduce the exact coordinates where the delivery point $B$ is located.
2. b) The company plans to open a new quick pickup point $C$ located further along the same straight line connecting $A$ and $M$. If this point $C$ must have an $x$-coordinate of $x = 11$, calculate the $y$-coordinate that must be assigned to it so that the three locations $A$, $M$, and $C$ are perfectly aligned. Justify the calculation by verifying that the vectors $\vec{AM}$ and $\vec{MC}$ have the same direction.

A warehouse robot moves along a straight conveyor belt. At a given moment, the system records it passing through point $A(1, 2)$ and, later, through position $B(4, 8)$. If a proximity sensor must be located on the x-axis, determine the exact coordinates of this sensor so that it is equidistant from points $A$ and $B$.

Check using vectors whether the quadrilateral with vertices $A(-4, -1)$, $B(2, -3)$, $C(4, 3)$ and $D(-2, 5)$ is a parallelogram. Then, calculate its exact perimetre, expressing the result using simplified surds.

Given the points $A(a, 1)$ and $B(2, a)$ depending on the real parameter $a$, find the possible values of this parameter such that the distance between $A$ and $B$ is exactly $\sqrt{5}$ units.

To install a new underground fibre-optic network, engineers must divide the straight line segment joining the operations centre $C(-5, 2)$ to the main node $N(7, -4)$ into four sections of equal length. Calculate the coordinates of the three intermediate connection points.

A point $P(x, y)$ satisfies the condition of having the same x-coordinate as y-coordinate. If it is known that the distance from $P$ to the point $M(2, 0)$ is equal to the distance from $P$ to the point $N(0, 4)$, set up the corresponding equation and determine the exact coordinates of point $P$.

In an aerospace positioning system, three satellites are located at coordinates $S_1(-2, k)$, $S_2(1, 2)$ and $S_3(k, -2)$. Determine the value of parameter $k$, knowing that, for calibration reasons, the three satellites are perfectly collinear and satisfy $k > 0$.

The points $A(-1, 3)$, $B(2, -1)$ and $C(x, y)$ form the vertices of a right-angled triangle where the right angle is at vertex $B$. Knowing that vertex $C$ is located on the y-axis, calculate the exact coordinates of point $C$ using Pythagoras' theorem.

Find the coordinates of the three vertices of a triangle $ABC$, knowing that the coordinates of the midpoints of its sides are $M(2, 1)$, $N(3, -2)$ and $P(0, -1)$. Set up the corresponding system of equations to solve it.

To check whether three forest fire lookout towers located at points $A(-3, -1)$, $B(1, 2)$, and $C(9, 8)$ are aligned (that is, if there is a continuous line of sight between the three), two park rangers propose different methods:

• Lucía's proposal: calculate the three distances $d(A, B)$, $d(B, C)$, and $d(A, C)$ using the Pythagorean theorem and check whether the metric relationship $d(A, B) + d(B, C) = d(A, C)$ holds.

• Marcos's proposal: calculate the vectors $\vec{AB}$ and $\vec{BC}$ and check whether their components are directly proportional.

1. a) Apply both methods explicitly step by step and determine whether the three towers lie on a straight line.
2. b) Critically compare both strategies: which of the two methods is more advantageous and reliable when the Cartesian coordinates do not yield exact square roots? Justify your answer considering the computational workload and numerical precision.

A regional waste management consortium in Spain needs to locate a new recycling plant to serve two municipalities: Aliseda, located at $A(2, 2)$, and El Pinar, at $B(8, 4)$ (coordinates expressed in kilometres). Due to heavy transport regulations, the plant must be built on a horizontal straight road with a fixed $y$-coordinate of $y = 9$, that is, at a point with generic coordinates $P(x, 9)$. To avoid disputes over transport costs, the plant must be situated at the exact same straight-line distance from both municipalities ($d(A, P) = d(B, P)$).

1. a) Set up the algebraic equation $[d(A, P)]^2 = [d(B, P)]^2$ in terms of the $x$-coordinate, expand it, and solve the resulting linear equation to find the exact position of the plant $P$.
2. b) In a preliminary meeting, a municipal representative proposed locating the facility at point $Q(5, 9)$, arguing that $x = 5$ is the arithmetic mean of the $x$-coordinates of both towns ($(2 + 8)/2 = 5$) and that it would therefore be equidistant from both. Calculate the numerical distances $d(A, Q)$ and $d(B, Q)$ to show that this intuitive proposal was incorrect, and explain geometrically why the plant must shift to the west ($x < 5$) to compensate for the difference in latitude between the two municipalities.