Exercises: Definitions

Classify the following quantities used in physics and geometry as scalars or vectors, briefly justifying your answer based on their definition:

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

• Wind velocity in a wind farm ($20 \text{ km/h}$ in a Northwest 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 programs the autopilot to make a rectilinear displacement defined by the vector $\vec{v} = (8, -3)$, determine the coordinates of the destination point $Q$.

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

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

To install a fiber 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 feet. Calculate the straight-line distance between both points.

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

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

Three lighting drones for a night show are located at coordinates $A(1, 2)$, $B(5, 3)$ and $C(6, 7)$. For the geometric figure they form in the sky to be a parallelogram $ABCD$, determine the coordinates of the position where the fourth drone $D$ must 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 that joins point $P(-2, 1)$ on the ground with point $Q(7, -5)$ on a platform. If they wish to place two intermediate supports that divide the staircase into three sections of equal length, calculate the coordinates where these supports must be anchored.

In a physics simulator, the wind direction is given by the vector $\vec{w} = (-3, 4)$. To correctly apply aerodynamic formulas, 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 modulus is exactly $10$.

Given the regular hexagon $ABCDEF$ in the figure, whose center is the origin of coordinates $O(0,0)$ and 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 robot makes two consecutive displacements. First, it moves from its base $O(0,0)$ to 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 it reaches point $Q$. Calculate the coordinates of point $Q$ and the total straight-line distance from the base $O$ to $Q$.

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 (they are parallel), opposite sense, and furthermore it is satisfied that the modulus 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 sides $AB, BC, CD$ and $DA$ respectively. Prove, using exclusively operations with vectors and coordinates of points, 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 relationship $2\vec{PA} + 3\vec{PB} = \vec{0}$ is satisfied.

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

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

The points $A(1, 1)$, $B(4, 2)$, $C(3, 5)$ and $D(0, 4)$ are given. Calculate the vectors that form 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 point $A(3, -1)$ with respect to point $P(1, 2)$. Set up the solution using the definition of a vector joining two points and indicate the geometric relationship between the vectors $\vec{AP}$ and $\vec{PA'}$.

Given the vectors $\vec{u} = (4, 0)$ and $\vec{v} = (0, 3)$. We wish to find a vector $\vec{w}$ that is the angle bisector of 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 angle bisector.