Vectors. 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 programs 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 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 meters. Calculate the straight-line distance between both points.

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 $k$.

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 $10$.

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.

Three consecutive vertices of an urban garden are located at coordinates $A(1, 2)$, $B(5, 3)$ and $C(6, 7)$, expressed in decameters. 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 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.

A support pillar is to be placed exactly at the center 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.

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.

Given the regular hexagon $ABCDEF$ in the figure, whose center 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).

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.

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.

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 $Q$.

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.

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).

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.