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 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 segment with endpoints $P(-4, 7)$ and $Q(8, -1)$.

Check whether 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 point symmetric to $A(1, 4)$ with respect to point $M(3, -2)$.

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

To obtain a unit vector with the same direction and sense as $\vec{v} = (-6, 8)$, Marcos reasons as follows on his exam:

"Since the components add up to $-6 + 8 = 2$, for the vector to have a length of $1$ I just need to divide each component by $2$. Thus, I obtain the vector $\vec{u} = \left(\frac{-6}{2}, \frac{8}{2}\right) = (-3, 4)$, which is already a unit vector".

1. a) Identify the two serious conceptual errors made by Marcos when calculating the magnitude and when defining what a unit vector is.
2. b) Analytically prove that the vector $(-3, 4)$ is not a unit vector by calculating its actual magnitude.
3. c) Correct the procedure and obtain the exact components of the true unit vector $\vec{u}$ with the same direction and sense as $\vec{v}$, verifying that its magnitude is exactly $1$.

Given the vectors $\vec{u} = (4, -1)$ and $\vec{v} = (-1, 3)$:

1. a) Before doing any calculations on paper, analyse the effect of multiplying $\vec{u}$ by $2$ and $\vec{v}$ by $3$, and make a reasoned estimate of which quadrant of the Cartesian plane the vector $\vec{w} = 2\vec{u} + 3\vec{v}$ will lie in. Justify your prediction by looking at the dominant signs of each component.
2. b) Visually compare the magnitudes of $2\vec{u}$ and $3\vec{v}$ and make an estimate: will the magnitude of $\vec{w}$ be greater than or less than $10$?
3. c) Accurately calculate the components of $\vec{w}$ and its exact magnitude $|\vec{w}|$.
4. d) Compare your calculations with the estimates from parts a) and b) and explain whether your initial conjectures were consistent with the result obtained.

A warehouse robot follows the following path from its base at the origin: first it moves according to vector $\vec{d_1} = (4,5; 2,1)$ m, then follows vector $\vec{d_2} = (-1,2; 3,4)$ m and finally moves according to $\vec{d_3} = (0; -1,5)$ m. Determine the final position vector of the robot and its straight-line distance from the base.

In video game design, 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 whether they have the same sense.

Express vector $\vec{w} = (1, 11)$ as a linear combination of 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 coordinates $A(-15, 20)$ and $B(35, -4)$ (in kilometres). A telecommunication antenna is to be installed exactly at the midpoint between them. Calculate the coordinates where the antenna should be placed and the length of the fibre 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 collinear.

Calculate the coordinates of a point $P$ that divides the 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 $\vec{AP} = 2\vec{PB}$ holds.

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 decametres). Calculate the perimetre of the plot.

A rescue drone must reach a position set by the total displacement vector $\vec{d}_{\text{total}} = (7, -1)$. To navigate, its system has two calibrated elementary movements: $\vec{u} = (3, 1)$ and $\vec{v} = (-1, 3)$. The autopilot must determine two scalars $a$ and $b$ such that $\vec{d}_{\text{total}} = a\vec{u} + b\vec{v}$.

1. a) Set up the system of two linear equations relating the horizontal and vertical components of the vectors.
2. b) Solve the system to find the exact values of the coefficients $a$ and $b$.
3. c) Interpret the geometric and physical meaning of the values and signs found for $a$ and $b$: has the direction of each basis vector been stretched, shrunk, or reversed? Justify your answer based on what the notes explain about scalar multiplication.

A DGT surveillance drone flies over a section of the A-3 motorway in the province of Cuenca. Its engines propel it with its own velocity vector relative to the air of $\vec{v}_p = (45, 60)\text{ km/h}$. However, the AEMET weather station in the area records a constant wind with velocity vector $\vec{v}_w = (-15, 20)\text{ km/h}$.

1. a) Model the resultant velocity of the drone relative to the ground as the vector sum $\vec{v}_r = \vec{v}_p + \vec{v}_w$ and determine its components.
2. b) Calculate the drone's own speed ($|\vec{v}_p|$) and the resultant speed over the ground ($|\vec{v}_r|$). Comparing both magnitudes, reason whether the wind is aiding or hindering the drone's overall progress.
3. c) If the drone maintains this flight for $15$ minutes ($0,25\text{ h}$), determine its net displacement vector $\vec{d} = 0,25\cdot \vec{v}_r$ and calculate how far in a straight line it will be from the starting point.

A rover on Mars starts from point $A(-3, 2)$ and makes a displacement described by vector $\vec{u} = (5, 4)$ to reach point $B$. From $B$, it moves towards point $C$ such that $\vec{BC} = -2\vec{u} + \vec{w}$, where $\vec{w} = (1, -6)$. Determine the coordinates of 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 points $A(-1, 3)$, $B(5, 1)$ and $C(6, 4)$. Calculate the coordinates of the fourth vertex $D$ knowing that vectors $\vec{AB}$ and $\vec{DC}$ are equivalent. Check your result by calculating the midpoint of both diagonals.

In particle physics, a proton beam follows a straight 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 points $A(1, 2)$, $B(3, -1)$ and $C(0, 4)$, find the coordinates of a point $D$ satisfying the vector equation $2\vec{AB} - 3\vec{CD} = \vec{v}$, knowing that vector $\vec{v}$ has coordinates $(-5, 2)$.

The centre of a regular hexagon is at the origin $O(0,0)$ 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.

Prove analytically that the quadrilateral with vertices $A(-2, 1)$, $B(4, 3)$, $C(6, -1)$, and $D(0, -3)$ is a parallelogram. To do this, verify that the opposite sides are equipollent vectors.

An aeroplane flies with its own velocity represented by vector $\vec{v_m} = (250, 150)$ km/h. The wind blows with a velocity $\vec{v_v} = (-30, 40)$ km/h. The actual velocity of the aeroplane relative to the ground is the sum of both vectors. The engine velocity 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 aeroplane is exactly $\vec{v_r} = (300, 0)$ km/h. 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 points $A(-3, 2)$, $B(1, 4)$ and $P$ is isosceles, with sides $AP$ and $BP$ having equal length. That is, $|\vec{AP}| = |\vec{BP}|$ must hold.

We want to divide the segment with endpoints $A(-4, 7)$ and $B(5, -2)$ into three exactly equal parts by introducing two interior points, $P$ and $Q$ (with $P$ located closer to $A$). In class, three proposals come up:

• Alba's proposal: Calculate the midpoint $M$ between $A$ and $B$, and then find the midpoint between $A$ and $M$.

• Bruno's proposal: Calculate the step vector $\vec{u} = \frac{1}{3}\vec{AB}$ and consecutively obtain $P = A + \vec{u}$ and $Q = P + \vec{u}$.

• Carmen's proposal: Set up two independent vector equations using proportions: $\vec{AP} = \frac{1}{2}\vec{PB}$ and $\vec{AQ} = 2\vec{QB}$, solving for the coordinates $(x, y)$ of each point.

1. a) Explain using a geometric argument why Alba's proposal is conceptually flawed and what division of the segment it would actually produce.
2. b) Compare Bruno's and Carmen's proposals: argue which of the two valid methods is more efficient, carries less risk of sign errors, and generalises better to divisions into $5$ or more parts.
3. c) Apply the most efficient method to calculate the exact coordinates of points $P$ and $Q$.

In the Cartesian plane, consider points $A(-2, 1)$, $B(4, 3)$, and $C(2, 7)$, vertices of a triangle.

1. a) Find the coordinates of the midpoint $M$ of side $BC$ and, using the vector symmetry relation $A' = 2M - A$, calculate the coordinates of point $A'$, the reflection of $A$ across $M$.
2. b) Calculate the components of vectors $\vec{AB}$, $\vec{AC}$, and $\vec{AA'}$, and check numerically whether the relation $\vec{AA'} = \vec{AB} + \vec{AC}$ holds.
3. c) Prove algebraically why the relation $\vec{AA'} = \vec{AB} + \vec{AC}$ always holds for any triangle in the plane by substituting the generic expressions $M = \frac{B + C}{2}$ and $A' = 2M - A$ into the vector $\vec{AA'} = A' - A$, without using numerical coordinates. From this result, justify what type of quadrilateral points $A$, $B$, $A'$, and $C$ form.