Exercises: Definitions

Classify the following quantities used in physics and geometry as scalar or vector quantities, briefly justifying your answer based on their definitions:

• Temperature of a data server ($45^\circ \text{C}$).

• Wind velocity at a wind farm ($20 \text{ km/h}$ in a North-West direction).

• Mass of an artificial satellite ($1200 \text{ kg}$).

• Displacement of a drone from its base to a delivery point.

A robotic arm on an assembly line moves in a two-dimensional Cartesian plane. If its initial position is the point $A(3, -2)$ and its final position after a movement is $B(-1, 5)$, determine the components of the displacement vector $\vec{AB}$.

A fishing boat is located at coordinates $P(-4, 2)$ on a nautical chart. If the captain programmes the autopilot to make a straight-line movement defined by the vector $\vec{v} = (8, -3)$, determine the coordinates of the destination point $Q$.

In a video game design, the velocity of a character is given by the vector $\vec{v} = (-6, 8)$ in pixels per frame. Calculate the magnitude of this vector to determine the character's actual speed on the screen.

Two model aircraft follow trajectories defined by the vectors $\vec{u} = (4, -6)$ and $\vec{v} = (-2, 3)$. Using the definition of parallel vectors, check whether both aircraft are flying in the same direction. Do they have the same sense?

To install a fibre optic cable between two buildings, surveyors mark the positions of the connections on a plan as $A(-3, 5)$ and $B(2, -7)$, with units in metres. Calculate the straight-line distance between both points.

A support pillar is to be placed exactly at the centre of a steel beam. If the ends of the beam are located at the coordinates $M(-1, 4)$ and $N(7, -2)$, determine the coordinates of the point where the pillar should go.

Observe the vectors represented in the following grid. Identify which pairs of vectors are equipollent (equal), which have the same magnitude but different direction, and which have the same direction but opposite sense.

When calculating the components and the length (magnitude) of the bound vector joining point $A(3, -2)$ with point $B(-1, 4)$, Marcos makes several mistakes on his exam. He writes the following:

1. Components: $\vec{AB} = (3 - (-1),\ -2 - 4) = (4,\ -6)$.

2. Magnitude: $|\vec{AB}| = \sqrt{4^2 - 6^2} = \sqrt{16 - 36} = \sqrt{-20}$ (concluding that the vector cannot exist because a distance cannot be the square root of a negative number).

1. a) Identify the two conceptual mistakes made by Marcos and explain why they contradict the definition of a bound vector and the derivation of the magnitude from the Pythagorean theorem.
2. b) Correct the procedure by calculating the correct components of $\vec{AB}$ and its exact and simplified magnitude.
3. c) Which vector did Marcos actually calculate in the first step?

A hiking trail in the Sierra de Guadarrama connects a shelter located at point $A(1, 2)$ with a viewpoint at point $B(6, 14)$ (coordinates expressed in kilometers).

1. a) Before calculating the exact distance, make a reasoned estimation: mentally deduce the length of the horizontal and vertical displacements (legs) and determine between which two integer values the magnitude $|\vec{AB}|$ must lie by applying the triangle inequality.
2. b) Calculate the exact components of vector $\vec{AB}$ and its magnitude using the Pythagorean theorem.
3. c) Check whether the obtained value matches your previous estimation and explain geometrically why the straight-line distance traveled can never exceed the sum of the absolute values of its components.

Three consecutive vertices of an urban garden are located at coordinates $A(1, 2)$, $B(5, 3)$ and $C(6, 7)$, expressed in decametres. The garden is intended to be in the shape of a parallelogram $ABCD$. Determine the coordinates where the fourth vertex $D$ should be located.

The force applied to an object is given by the vector $\vec{F} = (5, k)$ in Newtons. If a sensor indicates that the total magnitude of the force is $13 \text{ N}$, determine the possible values of the parameter $k$.

To lay out a straight road, engineers must ensure that three control points are perfectly aligned. The first two points are at $A(-1, 3)$ and $B(2, 5)$. If the third point has coordinate $y = 9$, find its coordinate $x$ using the concept of parallel vectors.

An architect designs a straight staircase connecting point $P(-2, 1)$ on the ground to point $Q(7, -5)$ on a platform. If they wish to place two intermediate supports dividing the staircase into three sections of equal length, calculate the coordinates where these supports should be anchored.

In a physics simulator, the wind direction is given by the vector $\vec{w} = (-3, 4)$. To apply the aerodynamic formulas correctly, the game engine needs a unit vector that maintains exactly the same direction and sense as $\vec{w}$. Calculate it.

Find a vector $\vec{v}$ that is parallel to the vector $\vec{u} = (1, -2)$, has the opposite sense to $\vec{u}$, and whose magnitude is exactly $10$.

Given the regular hexagon $ABCDEF$ in the figure, whose centre is the origin of coordinates $O(0,0)$ and whose vertex $A$ has coordinates $(2,0)$. Express the vectors $\vec{AC}$ and $\vec{AD}$ as a linear combination of the position vectors of the vertices (for example, by operating with points or adding known vectors from the figure).

A Martian exploration rover undergoes two consecutive displacements. First, it moves from its base $O(0,0)$ to the point $P$ defined by the vector $\vec{v_1} = (12, 5)$. Then, from $P$, it makes a second displacement $\vec{v_2} = (-4, 10)$ until reaching point $Q$. Calculate the coordinates of point $Q$ and the total straight-line distance from the base $O$ to $Q$.

At the control center of Adolfo Suárez Madrid-Barajas Airport, the position of two aircraft is monitored. At a given instant, a commercial airplane is at position $A(8, 15)$ and travels in a straight line to control point $B(20, 20)$ (coordinates in kilometers relative to the central radar $O(0, 0)$). Simultaneously, an emergency light aircraft departs from point $C(-6, -4)$ and performs an equipollent displacement to that of the commercial airplane up to a position $D$.

1. a) Write the position vector $\vec{OA}$ of the first airplane and obtain the components and magnitude of the displacement vector $\vec{AB}$. What does this magnitude represent geometrically?
2. b) Applying the definition of equipollent vectors, calculate the coordinates of the final position $D$ of the light aircraft.
3. c) The AEMET bulletin publishes: "Constant wind in the Barajas sector at $45\text{ km/h}$ along the North-South axis". Justify whether this statement defines a scalar quantity or a complete vector quantity, explaining what element is missing to determine the wind vector unambiguously.

We want to determine the coordinates of a point $Q(q_1, q_2)$ in the plane knowing that the bound vector $\vec{PQ}$, with initial point at $P(-4, 1)$, satisfies the following conditions:

1. Its magnitude is exactly equal to $13$.

2. Its vertical component $v_2$ is positive and its horizontal component $v_1$ exceeds said vertical component by $7$ units ($v_1 - v_2 = 7$).

1. a) State the equation for the magnitude from the Pythagorean definition, solve algebraically, and deduce the components of vector $\vec{PQ}$.
2. b) Using the relationship between endpoints and components of a bound vector, determine the exact coordinates of point $Q$.
3. c) Write the components of a vector $\vec{w}$ that has the same direction as $\vec{PQ}$, opposite sense, and exactly half its magnitude.

Determine the values of the parameters $a$ and $b$, knowing that the vectors $\vec{u} = (a, 3)$ and $\vec{v} = (-2, b)$ have the same direction (are parallel), opposite sense, and furthermore the magnitude of $\vec{u}$ is $5$.

Given any quadrilateral with vertices $A, B, C$ and $D$. Let $M, N, P$ and $Q$ be the midpoints of the sides $AB, BC, CD$ and $DA$ respectively. Prove, using exclusively vector operations and point coordinates, that the quadrilateral $MNPQ$ is always a parallelogram.

Given the points $A(1, -2)$ and $B(6, 3)$, find the coordinates of a point $P$ located on the line segment joining them, such that the vector relation $2\vec{PA} + 3\vec{PB} = \vec{0}$ holds.

Find the coordinates of all points located on the x-axis ($X$ axis) whose distance to point $A(-2, 3)$ is exactly equal to $5$ units.

A boat attempts to cross a river perpendicular to the banks with its own velocity given by the vector $\vec{v_b} = (0, 12)$ km/h. However, the river current exerts a constant velocity represented by the vector $\vec{v_c} = (5, 0)$ km/h. Calculate the actual velocity vector of the boat relative to an observer on the bank and determine its actual speed (magnitude of the resultant vector).

The points $A(1, 1)$, $B(4, 2)$, $C(3, 5)$ and $D(0, 4)$ are given. Calculate the vectors forming the sides of the quadrilateral $ABCD$ and the lengths of its diagonals. Based on these calculations (and without using the scalar product), justify whether the figure is a rhombus, a square, a rectangle or none of these regular figures.

Find the coordinates of the symmetric point $A'$ of the point $A(3, -1)$ with respect to the point $P(1, 2)$. Set up the solution using the definition of a vector joining two points and state the geometric relationship between the vectors $\vec{AP}$ and $\vec{PA'}$.

Given the vectors $\vec{u} = (4, 0)$ and $\vec{v} = (0, 3)$. It is desired to find a vector $\vec{w}$ that bisects the angle formed by $\vec{u}$ and $\vec{v}$. Knowing that the sum of two unit vectors results in a vector in the direction of the angle bisector, calculate the exact coordinates of a direction vector $\vec{w}$ of this bisector.

To find the coordinates of the fourth vertex $D$ of a parallelogram whose other three consecutive vertices are $A(-2, 1)$, $B(3, -2)$, and $C(7, 4)$, three 1st year Bachillerato students propose different strategies:

• Elena proposes Method 1: using the equipollence of free vectors on opposite sides, imposing that $\vec{AB} = \vec{DC}$.

• Guille proposes Method 2: calculating the position vector of the midpoint of diagonal $AC$, $\vec{OM} = \dfrac{\vec{OA}+\vec{OC}}{2}$, and setting it equal to $\dfrac{\vec{OB}+\vec{OD}}{2}$ to solve for $D$.

• Sara proposes Method 3: setting up a system of two quadratic equations by equating the magnitudes of the opposite sides: $|\vec{AB}| = |\vec{CD}|$ and $|\vec{BC}| = |\vec{DA}|$.

1. a) Apply both Method 1 and Method 2 to find the coordinates of $D$, verifying that both methods lead to the same result.
2. b) Explain in detail why Sara's Method 3 is flawed and inadequate: what other non-parallelogram geometric configurations satisfy those distance equalities, and what algebraic problem arises when solving that system?
3. c) Argue which of the first two methods is conceptually more direct and generalizable to three dimensions without the need to resort to distances or angles.

A Salvamento Marítimo vessel based in Tarifa must anchor two intermediate beacons, $P$ and $Q$, along the straight segment joining two main buoys located at $A(1, 3)$ and $B(19, 27)$ (coordinates in nautical miles referred to a coastal station $O(0, 0)$). The intermediate beacons must divide the oriented path $\vec{AB}$ into three equal sections of the same sense, satisfying $\vec{AP} = \vec{PQ} = \vec{QB}$.

1. a) Formally prove, using the relationship between points and position vectors ($\vec{AB} = \vec{OB} - \vec{OA}$), that the position vector of beacon $P$ is given by $\vec{OP} = \dfrac{2\vec{OA}+\vec{OB}}{3}$, and obtain with reasoning the analogous expression for $\vec{OQ}$.
2. b) Calculate the exact coordinates of $P$ and $Q$, and calculate the total distance between $A$ and $B$.
3. c) A rescue helicopter carries out a reconnaissance flight according to the vector $\vec{CD} = (12, 16)$. Justify with mathematical rigor whether vectors $\vec{AB}$ and $\vec{CD}$ have the same direction, the same sense, and whether they are equipollent or not.