Exercises: Vectors in two dimensions

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

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

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

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

Look at the vectors $\vec{u}$ and $\vec{v}$ represented on the grid. Write down their components and analytically calculate 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 (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 $10$.

In an Analytic Geometry test, Marcos needs to calculate the components of the vector $\vec{AB}$ connecting the initial point $A(3, -2)$ with the terminal point $B(-1, 4)$, as well as its exact magnitude. On his exam paper, he writes the following:

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

2) $|\vec{AB}| = \sqrt{4^2 - 6^2} = \sqrt{16 - 36} = \sqrt{-20}$

After observing the result, Marcos concludes that “the vector does not exist in the real plane because the square root of a negative number is not a real number”.

1. a) Identify and explain in detail the two conceptual errors that Marcos made in his approach.
2. b) Correct the problem step by step: find the correct components of $\vec{AB}$, calculate its exact magnitude using radicals, and also give its approximate value to one decimal place.

A forest ranger walks along a straight path from the watch base located at point $P(2, 1)$ to a viewpoint located at $Q(9, 5)$ (coordinates in kilometres).

1. a) Before calculating the exact magnitude, observe the displacement that occurs along the horizontal axis ($x$) and the vertical axis ($y$). Keeping in mind that the hypotenuse of a right triangle is always greater than the larger of its legs and smaller than the sum of both, and comparing the sum of the squares of both displacements with the nearest perfect squares, make a reasoned estimate and bound between which two consecutive integers the total distance covered (the magnitude of the vector $\vec{PQ}$) must lie.
2. b) Calculate the components of the vector $\vec{PQ}$ and find its exact magnitude in the form of a square root.
3. c) Obtain the decimal approximation of this magnitude to two decimal places, check whether your estimate from section a) was correct, and reason whether the obtained result is consistent with the physical situation.

A delivery drone flies with its own velocity represented by the vector $\vec{v}_d = (12, 5)$ km/h. At that instant, the wind blows with a velocity $\vec{v}_v = (-3, 2)$ km/h. 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 metres). Calculate the coordinates of the fourth vertex $D$ using vector operations.

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

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 $\vec{R} = (4, 3)$ 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 $A(-15, 20)$ to a reservoir $B(45, -12)$ (coordinates in metres). A control valve is to be installed exactly at the midpoint of the route. Calculate the coordinates where the valve will be placed and the total length of the pipe.

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

A laser beam originates 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 symmetric point of $F$ with respect to the mirror and write down the components of the vector that defines the reflected path.

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

The Maritime Rescue coordination centre in Tarifa is monitoring a fuel slick drifting in the Strait of Gibraltar. Data indicate that the spill is moving in the direction and sense of the vector $\vec{v} = (6, -2)$ (in nautical miles per hour). To contain the oil slick, a response vessel must deploy a floating barrier in a direction strictly perpendicular to the advance of the spill.

1. a) Model the problem and calculate the value of the coordinate $x$ so that the vector $\vec{u} = (x, 12)$ is perpendicular to the direction of the spill $\vec{v}$, applying the dot product condition.
2. b) The vessel has barrier sections totaling a length of $15$ nautical miles. Calculate the magnitude of the vector $\vec{u}$ obtained in the previous section and determine whether the available barrier is shorter than, equal to, or longer than $|\vec{u}|$.
3. c) By what positive real number $k$ must the vector $\vec{u}$ be multiplied to obtain a vector with the same direction and sense, but whose length exactly matches the $15$ nautical miles of the available barrier?

In the design of a logo with a technical aesthetic, two straight strokes are represented by the vectors $\vec{u}$ and $\vec{v}$. It is known that the vector $\vec{u}$ has its initial point at $A(-2, 1)$ and its terminal point at $B(2, 4)$. The second vector has components $\vec{v} = (k, -8)$, its initial point is located at $C(5, -1)$, and it must necessarily be perpendicular to $\vec{u}$.

1. a) Set up the corresponding equation and calculate the value of the real parameter $k$ so that both vectors are perpendicular.
2. b) With the value of $k$ now determined, find the coordinates of point $D$, which is the terminal point of the vector $\vec{v}$.
3. c) Compare the lengths of both strokes: calculate the magnitudes $|\vec{u}|$ and $|\vec{v}|$ and state the exact proportionality ratio between the length of the vector $\vec{v}$ and that of the vector $\vec{u}$.

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

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

Consider the vectors depending 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 area at the points $A(-2, -1)$, $B(4, 1)$, $C(5, 5)$ and $D(-1, 3)$. First, show using vectors that the figure formed is a parallelogram. Next, calculate the scalar product of its diagonals to check whether 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. 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 $\vec{R} = \vec{F}_1 + \vec{F}_2$ is in the direction of the positive x-axis and has 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 plot. Next, find the coordinates of the exact centre of the square.

The centroid $G$ (centre of gravity) of a triangle is always located at a distance of $\frac{2}{3}$ from any vertex along the line 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 vector $\vec{AM}$, calculate the exact coordinates of the centroid $G$.

An agricultural cooperative is going to acquire a triangular plot whose vertices on the cadastral map are $A(-2, 1)$, $B(2, 4)$, and $C(-1, 8)$ (coordinates expressed in hectometres). Before executing the deed of sale, the technicians must certify whether the plot contains a right angle ($90^\circ$) to be able to locate a tool shed at that corner.

Two technicians propose different approaches:

• Metric method (Marta): Find the lengths of the three sides using the magnitudes $|\vec{AB}|$, $|\vec{BC}|$, and $|\vec{AC}|$, and check whether they satisfy the Pythagorean theorem equality.

• Vector method (Carlos): Calculate the components of the vectors of the sides meeting at each vertex and compute their dot products to check if any of them equals zero.

1. a) Solve the problem using both methods completely and confirm at which vertex the right angle is located.
2. b) Critically evaluate both procedures: which of the two methods is more efficient and direct from the standpoint of algebraic calculation? Justify your answer by analysing the need to calculate square roots and the risk of carrying over rounding errors in each case.

An architecture studio designs a rectangular pergola $ABCD$ in a park in Zaragoza. In the project, the coordinates of three of its vertices have been set: $A(1, 2)$, $B(5, 5)$, and $C(2, 9)$.

1. a) Analytically prove that the adjacent sides represented by the vectors $\vec{AB}$ and $\vec{BC}$ form a right angle using the dot product.
2. b) To locate the fourth vertex $D(x, y)$ such that the figure is a rectangular parallelogram, the vector equality $\vec{AD} = \vec{BC}$ must hold. Deduce the exact coordinates of point $D$ from it.
3. c) Two steel cables will be installed as diagonal tensioners joining opposite vertices: one between $A$ and $C$, and another between $B$ and $D$. Calculate the magnitudes $|\vec{AC}|$ and $|\vec{BD}|$ and analytically prove that both diagonals have the same length.
4. d) Calculate the dot product of the diagonal vectors $\vec{AC} \cdot \vec{BD}$ and interpret the result geometrically: what additional property does this rectangle have regarding its sides?