Exercises: Vectors in the plane

Given the points $A(2, -1)$ and $B(5, 3)$, calculate the components of the vector $\vec{AB}$ and its magnitude.

Consider the vectors $\vec{u} = (3, -2)$ and $\vec{v} = (-1, 4)$. Calculate the components of the resultant vector of the operation $\vec{w} = 2\vec{u} - 3\vec{v}$.

Determine the coordinates of the midpoint $M$ of the line segment whose endpoints are the points $P(-4, 7)$ and $Q(2, -3)$.

The point $M(1, 2)$ is the midpoint of the line segment $AB$. If we know the coordinates of the endpoint $A(-3, 5)$, find the coordinates of the endpoint $B$.

Observe the vectors $\vec{u}$ and $\vec{v}$ represented on the grid. Write down their components and calculate analytically the coordinates of the sum vector $\vec{s} = \vec{u} + \vec{v}$.

State whether the vectors $\vec{a} = (6, -9)$ and $\vec{b} = (-2, 3)$ are parallel (they have the same direction). Justify your answer mathematically.

Calculate the scalar product of the vectors $\vec{p} = (4, 5)$ and $\vec{q} = (-2, 3)$. From the result, state whether they are perpendicular.

Find the possible values of $x$ so that the vector $\vec{v} = (x, -8)$ has a magnitude exactly equal to $10$.

A delivery drone flies with its own velocity represented by the vector $\vec{v}_d = (12, 5)$ mph. At that moment, the wind blows with a velocity of $\vec{v}_v = (-3, 2)$ mph. Calculate the actual velocity vector of the drone and its speed (the magnitude of the resultant vector).

To install a solar panel in the shape of a parallelogram $ABCD$, the technicians have fixed three of its vertices at the coordinates $A(1, 1)$, $B(5, 2)$ and $C(6, 6)$ (in feet). Calculate the coordinates of the fourth vertex $D$ using vector operations.

In the design of a bridge, three pillars must be perfectly aligned. On the architect's plan, the bases of the pillars are located at the coordinates $P(-2, 1)$, $Q(2, 3)$ and $R(8, k)$. Determine the value of $k$ so that the alignment is correct.

In a physics experiment, two forces $\vec{F}_1 = (2, -1)$ N and $\vec{F}_2 = (-1, 3)$ N are applied. We want to obtain a resultant force of $\vec{R} = (4, 3)$ N by combining the first two using a system that multiplies their intensity. Find the values of the scalars $a$ and $b$ such that $\vec{R} = a\vec{F}_1 + b\vec{F}_2$.

A straight pipe connects a pumping station $A(-15, 20)$ with a tank $B(45, -12)$ (coordinates in feet). A control valve is to be installed exactly at the midpoint of the route. Calculate the coordinates where the valve will be located and the total length of the pipe.

On a billiard table, a ball is struck and follows the trajectory of the vector $\vec{u} = (4, 3)$. After bouncing off the cushion, its new trajectory is given by the vector $\vec{v} = (-3, k)$. Calculate the value of $k$ so that the rebound trajectory forms an angle of $90^\circ$ with the arrival trajectory.

A laser beam starts from a source located at $F(-3, -2)$ and reflects off a flat mirror that coincides with the x-axis. If the exact point of impact on the mirror is $I(1, 0)$, find the coordinates of the symmetrical point of $F$ with respect to the mirror and write down the components of the vector that defines the reflected trajectory.

Find out if the triangle formed by the telecommunications antennas located at the coordinates $A(1, 2)$, $B(4, 6)$ and $C(9, -1)$ is a right triangle. Use the scalar product of the vectors that form its sides to justify your answer.

A steel beam whose ends are at the coordinates $A(-4, 2)$ and $B(8, 14)$ must be cut into three identical parts to manufacture supports. Calculate the coordinates of the two cutting points $P$ and $Q$ that divide the line segment $AB$ into three parts of equal length.

Determine the coordinates of a point $P$ located on the y-axis that is at the same distance from the points $A(2, 5)$ and $B(-4, 1)$.

Consider the vectors dependent on a parameter $\vec{u} = (m, 2)$ and $\vec{v} = (3, m-1)$. Find the values of the parameter $m$ in two different geometric cases:

1. a) So that $\vec{u}$ and $\vec{v}$ are perpendicular.
2. b) So that $\vec{u}$ and $\vec{v}$ are parallel.

Four buoys delimit a marine reserve zone at the points $A(-2, -1)$, $B(4, 1)$, $C(5, 5)$ and $D(-1, 3)$. Show first, using vectors, that the figure formed is a parallelogram. Then, calculate the scalar product of its diagonals to check if it is specifically a rhombus.

Find the components of a vector $\vec{w}$ that is perpendicular to the vector $\vec{u} = (-5, 12)$ and whose magnitude is exactly half the magnitude of $\vec{u}$. How many mathematical solutions exist for $\vec{w}$? Write them all down.

Two tugboats pull a broken-down ship located at the origin of coordinates. The first tugboat exerts a force defined by the vector $\vec{F}_1 = (6, 8)$ kN. The second tugboat exerts a force $\vec{F}_2$. Knowing that the resultant force $\vec{R} = \vec{F}_1 + \vec{F}_2$ has the direction of the positive x-axis and a magnitude of $12$ kN, determine the components of the vector $\vec{F}_2$ and calculate its magnitude.

The points $A(-1, 3)$ and $B(5, -5)$ represent two opposite vertices of a square-shaped plot of land $ABCD$. Calculate the length of the diagonal of the square and the total area of the land. Then, find the coordinates of the exact center of the square.

The centroid $G$ (center of gravity) of a triangle is always located at a distance of $\frac{2}{3}$ from any vertex to the midpoint of the opposite side. If the vertices of a triangular metal plate are $A(1, 1)$, $B(7, 3)$ and $C(4, 8)$, first find the midpoint $M$ of the side $BC$. Then, using the vector $\vec{AM}$, calculate the exact coordinates of the centroid $G$.