Exercises: Operations with vectors

Given the points $A(3, -2)$ and $B(5, 4)$, calculate the coordinates of the vector $\vec{AB}$ and the vector $\vec{BA}$.

Calculate the magnitude of the vector $\vec{v} = (-6, 8)$.

Given the vectors $\vec{u} = (4, -1)$ and $\vec{v} = (-2, 5)$, calculate the coordinates of the sum vector $\vec{u} + \vec{v}$ and the difference vector $\vec{u} - \vec{v}$.

Given the vector $\vec{w} = (3, -7)$, calculate the coordinates of the vectors $2\vec{w}$ and $-\frac{1}{2}\vec{w}$.

Calculate the coordinates of the midpoint of the line segment with endpoints $P(-4, 7)$ and $Q(8, -1)$.

Check if the vectors $\vec{a} = (2, -5)$ and $\vec{b} = (-6, 15)$ have the same direction, that is, if they are parallel.

Find the coordinates of the symmetrical point of $A(1, 4)$ with respect to the point $M(3, -2)$.

The vector $\vec{v} = (5, 2)$ has its origin at the point $A(-3, 1)$. Determine the coordinates of its endpoint $B$.

A delivery drone makes the following journey from its base at the origin of coordinates: first it moves according to the vector $\vec{d_1} = (4.5, 2.1)$ miles, then it follows the vector $\vec{d_2} = (-1.2, 3.4)$ miles and finally it advances according to $\vec{d_3} = (0, -1.5)$ miles. Determine the final position vector of the drone and at what straight-line distance it is from the base.

In the design of a video game, the movement of a character is given by the vector $\vec{u} = (x, 12)$, and that of a projectile by the vector $\vec{v} = (5, -4)$. Calculate the value of $x$ so that both vectors have the same direction and find out if they have the same sense.

Express the vector $\vec{w} = (1, 11)$ as a linear combination of the vectors $\vec{u} = (2, 3)$ and $\vec{v} = (-1, 5)$. That is, find the values of $a$ and $b$ such that $\vec{w} = a\vec{u} + b\vec{v}$.

Two cities are located on a map at the coordinates $A(-15, 20)$ and $B(35, -4)$ (in miles). A telecommunications antenna is to be installed exactly at the midpoint between them. Calculate the coordinates where the antenna must be located and the length of the fiber-optic cable needed to connect it in a straight line with city $A$.

In a civil engineering project, three pillars must be aligned. On the plan, their positions are $P(1, -2)$, $Q(4, 2)$ and $R(10, k)$. Find the value of $k$ so that the three pillars are perfectly aligned.

Calculate the coordinates of a point $P$ that divides the line segment with endpoints $A(-2, 5)$ and $B(7, -1)$ such that the distance from $A$ to $P$ is twice the distance from $P$ to $B$. That is, the vector relationship $\vec{AP} = 2\vec{PB}$ is satisfied.

Two forces represented by the vectors $\vec{F_1} = (15, -8)$ N and $\vec{F_2} = (-5, 12)$ N act on a body. Calculate the resultant force vector $\vec{R} = \vec{F_1} + \vec{F_2}$ and determine the magnitude of this resultant force.

The vertices of a triangular plot are $A(0, 0)$, $B(8, 6)$ and $C(12, -3)$ (measured in decameters). Calculate the perimeter of the plot.

An exploration robot on Mars starts from the point $A(-3, 2)$ and makes a displacement described by the vector $\vec{u} = (5, 4)$ until it reaches a point $B$. From $B$, it moves towards a point $C$ such that $\vec{BC} = -2\vec{u} + \vec{w}$, where $\vec{w} = (1, -6)$. Determine the coordinates of the points $B$ and $C$, and calculate the magnitude of the total displacement vector $\vec{AC}$.

Three of the vertices of a solar panel in the shape of a parallelogram $ABCD$ are located at the points $A(-1, 3)$, $B(5, 1)$ and $C(6, 4)$. Calculate the coordinates of the fourth vertex $D$ knowing that the vectors $\vec{AB}$ and $\vec{DC}$ are equal. Check your result by calculating the midpoint of both diagonals.

In particle physics, a proton beam follows a rectilinear trajectory with direction vector $\vec{v} = (3, -4)$. Find a unit vector (of magnitude 1) that has the same direction and sense as $\vec{v}$, and another vector that has the same direction, opposite sense and magnitude 10.

Given the points $A(1, 2)$, $B(3, -1)$ and $C(0, 4)$, find the coordinates of a point $D$ that satisfies the vector equation $2\vec{AB} - 3\vec{CD} = \vec{v}$, knowing that the vector $\vec{v}$ has coordinates $(-5, 2)$.

The center of a regular hexagon is located at the origin of coordinates $O(0,0)$ and one of its vertices is the point $A(4, 0)$. Using the properties of vectors and knowing that the hexagon can be divided into 6 identical equilateral triangles, determine the coordinates of the other five vertices.

Show analytically that the quadrilateral with vertices $A(-2, 1)$, $B(4, 3)$, $C(6, -1)$ and $D(0, -3)$ is a rectangle. To do this, check that the opposite sides are equal vectors and that the diagonals have exactly the same length.

An airplane flies with its own velocity represented by the vector $\vec{v_m} = (250, 150)$ mph. The wind blows with a velocity of $\vec{v_v} = (-30, 40)$ mph. The actual velocity of the airplane with respect to the ground is the sum of both vectors. The velocity of the engines is to be modified by adding a correction vector $\vec{c} = a\vec{u} + b\vec{w}$, where $\vec{u} = (1, 0)$ and $\vec{w} = (0, 1)$, so that the final actual velocity of the airplane is exactly $\vec{v_r} = (300, 0)$ mph. Set up the vector equation and find the values of $a$ and $b$.

Find the coordinates of a point $P$ located on the y-axis such that the triangle formed by the points $A(-3, 2)$, $B(1, 4)$ and $P$ is isosceles, with the sides $AP$ and $BP$ being of equal length. That is, it must be satisfied that $|\vec{AP}| = |\vec{BP}|$.