Exercises: Vectors

A student calculated the components of the vector from point $A(2, -5)$ to point $B(-1, 4)$ by carrying out the operation $A - B$, obtaining the vector $(3, -9)$. State the conceptual error made and calculate the correct components of vector $\vec{AB}$.

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

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

Calculate the coordinates of the midpoint of the line segment with endpoints $P(7, -2)$ and $Q(-3, 6)$.

Find the distance between the points $C(-1, -4)$ and $D(5, 4)$.

Check analytically whether vectors $\vec{a} = (6, -9)$ and $\vec{b} = (-4, 6)$ are parallel (linearly dependent).

Calculate the scalar product of vectors $\vec{p} = (5, 2)$ and $\vec{q} = (-3, 4)$.

Given point $A(1, 4)$ and the midpoint $M(3, -1)$ of line segment $AB$, find the coordinates of endpoint $B$.

Determine, using the scalar product, whether vectors $\vec{m} = (4, 6)$ and $\vec{n} = (-3, 2)$ are perpendicular.

Marcos wants to check whether the trajectory of a drone represented by the vector $\vec{u} = (4, -3)$ and a wind gust $\vec{v} = (6, 8)$ form a right angle (that is, whether they are perpendicular). In his exam, he writes the following:

"$\vec{u} \cdot \vec{v} = (4 \cdot 6, -3 \cdot 8) = (24, -24)$. Since the result is not $(0, 0)$, the vectors are not perpendicular."

1. a) Identify and explain the two conceptual errors Marcos made: one regarding the nature of the dot product and the other regarding the final result for perpendicularity.
2. b) Perform the correct calculation of the dot product and determine, with reasoning, whether the directions of the drone and the wind are perpendicular or not.

A small fishing boat is sailing near the Delta del Ebro. Its engine provides a velocity described by the vector $\vec{v}_m = (12, 5)$ in knots (nautical miles per hour, where the horizontal axis points east and the vertical axis points north). At that moment, a Mestral wind recorded by AEMET sets in, exerting on the boat an additional velocity given by the vector $\vec{v}_w = (4, -5)$ knots.

1. a) Without carrying out complex operations, estimate with reasoning what will happen to the north-south displacement of the boat when both effects combine, and whether the final speed of the fishing boat will be greater or less than the $13$ knots provided solely by the engine.
2. b) Verify your estimate by calculating the resulting velocity vector $\vec{v}_r = \vec{v}_m + \vec{v}_w$ and its exact magnitude. Explain the physical meaning of the obtained result with respect to your prediction.

Two aeroplanes depart from the same point along straight trajectories described by vectors $\vec{v_1} = (120, 160)$ and $\vec{v_2} = (200, -150)$ (in km/h). Calculate the angle formed by their trajectories.

Three forces represented by vectors $\vec{F_1} = (15, -10)$, $\vec{F_2} = (-5, 25)$ and $\vec{F_3} = (-4, -3)$ in newtons act on a point object. Calculate the vector of the resultant force and determine its magnitude.

Find the value (or values) of $k$ such that the magnitude of vector $\vec{v} = (k, -8)$ is exactly $10$.

A drone moves in a straight line from point $A(-4, 2)$ to point $B(8, 14)$ on a coordinate map in metres. If it stops exactly one third of the way from the starting point $A$, at what exact coordinates does it stop?

For two beams of a steel structure to cross at right angles, their direction vectors must be perpendicular. If the first beam follows the direction $\vec{u} = (k-1, 3)$ and the second $\vec{v} = (4, k)$, calculate the value of parameter $k$.

On a football pitch modelled on a Cartesian plane, a player is at position $J(-12, 5)$ and the ball is at $B(18, -11)$ (measurements in metres). Calculate the straight-line distance between the player and the ball, and the coordinates of the exact point where the player will have travelled half the way to the ball.

Three telecommunication satellites are located at orbital positions $S_1(-2, 3)$, $S_2(1, 7)$ and $S_3(7, 15)$. Check analytically whether they are collinear along the same trajectory.

In a video game engine, a character moves in the direction of vector $\vec{d} = (3, -4)$. The programmer needs to generate a vector perpendicular to the character's trajectory, but with a magnitude of $15$ units. Find the components of the two possible vectors that satisfy this condition.

In a nautical simulator, you need to program two propulsion vectors, $\vec{a}$ and $\vec{b}$, that interact with an ocean current given by the vector $\vec{u} = (4, -2)$. You must determine the numerical components of $\vec{a}$ and $\vec{b}$ satisfying these three conditions simultaneously:

1) None of the components of the vectors can be equal to zero.

2) Vector $\vec{a}$ must be perpendicular to current $\vec{u}$.

3) Vector $\vec{b}$ must have the same direction as $\vec{a}$, but opposite sense and a magnitude that is exactly three times the magnitude of $\vec{a}$.

Write the components of a pair of vectors that satisfies these conditions and justify step by step how you deduced each one from the geometric properties of vectors.

An eco-friendly delivery van delivers packages in the metropolitan area of Madrid starting from the logistics center in Coslada. Its first trip is given by the vector $\vec{d}_1 = (6, 8)$ (in kilometers, where the $X$-axis represents eastward movement and the $Y$-axis northward). From there, it makes a second displacement $\vec{d}_2 = (-2, 4)\text{ km}$ to the last delivery.

1. a) Model the total trip by determining the resulting displacement vector $\vec{d}_{\text{total}}$ from the center to the final point and calculate what straight-line distance it is from the base.
2. b) To return directly to the center without stops, what return vector $\vec{d}_{\text{regreso}}$ must the navigator display? Justify what relationship its magnitude, direction, and sense have with respect to $\vec{d}_{\text{total}}$.
3. c) A surveillance drone departs from the center following the route marked by the vector $\vec{w} = (6, -2)$. Determine using the dot product whether the trajectory of the drone is perpendicular to the direct return route of the van.

Four stakes mark the vertices of a parallelogram-shaped plot $ABCD$. The coordinates of three of them are known: $A(-3, -1)$, $B(2, -2)$ and $C(5, 3)$. Calculate the coordinates of the fourth stake $D$, knowing that it is the vertex opposite $B$.

A laser cutting machine needs to cut out a triangular piece with vertices at $P(-4, 2)$, $Q(2, 6)$ and $R(4, -1)$ (in cm). Prove analytically what type of triangle it is (equilateral, isosceles or scalene) and calculate the exact perimetre of the piece.

Determine the value of $m$ so that the vectors $\vec{u} = (m, 1)$ and $\vec{v} = (2, -3)$ form an exact angle of $135^\circ$.

In a physics experiment, the displacement of a particle is given by the vector $\vec{w} = (11, -2)$. Express this vector as a linear combination of the fluid flow vectors $\vec{u} = (3, 2)$ and $\vec{v} = (1, -4)$. That is, find the scalars $x$ and $y$ such that $\vec{w} = x\vec{u} + y\vec{v}$.

A Wi-Fi router needs to be installed in a straight corridor that coincides with the x-axis. The router must be at exactly the same distance from two computers located at points $A(-2, 5)$ and $B(4, 1)$. Calculate the exact coordinates where the router should be placed.

Given the triangle with vertices $A(-1, 0)$, $B(3, 2)$ and $C(1, 6)$, prove using the scalar product that it is a right-angled triangle and then calculate its area.

A suspension bridge has a straight cable connecting anchor point $A(-6, 8)$ to tower $B(10, -4)$ (in metres). Three lights are to be placed, dividing the cable into four sections of equal length. Calculate the coordinates of the three light points.

A billiard ball hits the cushion of the table. The incoming trajectory is defined by vector $\vec{v_i} = (4, -3)$. If the cushion is parallel to the x-axis, the outgoing trajectory vector $\vec{v_f}$ will have the same x-component but the opposite y-component. Calculate the exact angle formed by the incoming and outgoing trajectory vectors.

To check whether the triangle with vertices $A(-3, 1)$, $B(1, 4)$, and $C(-2, 8)$ has a right angle at vertex $B$, two students propose different strategies:

• Carla's strategy: Calculate the lengths of the three sides using their magnitudes ($|\vec{AB}|$, $|\vec{BC}|$, and $|\vec{AC}|$) and check whether the Pythagorean theorem holds ($|\vec{AC}|^2 = |\vec{AB}|^2 + |\vec{BC}|^2$).

• Hugo's strategy: Obtain only the vectors of the sides meeting at $B$, that is, $\vec{BA}$ and $\vec{BC}$, and calculate their dot product to see if it equals zero.

1. a) Apply both strategies step by step to solve the geometric question.
2. b) Compare both procedures: how many intermediate operations does each require? Justify which of the two strategies is more efficient, faster, and less prone to calculation errors when working with vectors.

The Maritime Rescue service in the Strait of Gibraltar monitors on its radar (with distances in nautical miles) a drifting dinghy that departed from point $A(2, 3)$. Due to the ocean current, the boat drifts each hour according to the vector $\vec{c} = (3, 1)$.

1. a) Model using vector operations the position $P$ where the boat will be after $3$ hours of uninterrupted drifting.
2. b) From the coastal station located at $B(2, -6)$, a rescue vessel departs at the moment of the alarm to intercept the boat at point $P$ exactly after those $3$ hours. Determine the components of the displacement vector $\vec{s}$ that the rescue vessel must complete.
3. c) Knowing that the maximum speed of the rescue boat is $5$ knots ($5$ nautical miles per hour), calculate the magnitude of $\vec{s}$ and justify mathematically whether the vessel has enough power to arrive on time at the rendezvous. In what simplified direction and sense should the crew set course?