Exercises: Vectors and vector spaces

Carlos has to find the coordinates of the midpoint $M$ of the segment with endpoints $A(1,\; -2,\; 4)$ and $B(5,\; 2,\; -2)$. To solve it, he writes on his exam:

Therefore, he concludes that the midpoint is $M(2,\; 2,\; -3)$.

1. a) Explain in detail what Carlos's conceptual error is. What does the vector $\frac{1}{2}\vec{AB}$ represent geometrically, and why do its components not coincide with the coordinates of the midpoint relative to the origin?
2. b) Show how Carlos could have reached the correct midpoint using his vector $\frac{1}{2}\vec{AB}$ by adding it to the appropriate position vector.
3. c) Correct the calculation by applying the direct midpoint formula $M = \frac{A+B}{2}$ and verify that both approaches lead to the same result.

Consider the space vector $\vec{v} = (2,\; -6,\; 9)$.

1. a) Without calculating square roots, justify why the magnitude of any non-zero vector $\vec{u} = (u_1,\; u_2,\; u_3)$ is always greater than or equal to the absolute value of any of its components ($|\vec{u}| \ge |u_i|$, with strict inequality if it has at least two non-zero components) and less than or equal to the sum of the absolute values of all of them ($|\vec{u}| \le |u_1| + |u_2| + |u_3|$).
2. b) Use the reasoning above to bound the value of $|\vec{v}|$ between two integers, estimating an interval in which its length must necessarily lie.
3. c) Calculate the exact magnitude of $\vec{v}$ by applying the three-dimensional formula and check whether your initial estimate was correct.

A delivery drone moves in three-dimensional space following paths defined by vectors. Its basic thrusters allow it to perform three fundamental movements defined by the vectors $\vec{u} = (1, -1, 0)$, $\vec{v} = (0, 2, 1)$ and $\vec{w} = (2, 0, 3)$. If the drone needs to make an urgent delivery from the control centre to a customer following the resulting displacement vector $\vec{d} = (5, 1, 10)$, calculate the coefficients $a, b$ and $c$ such that the movement can be expressed as a linear combination of the thrusters: $\vec{d} = a\vec{u} + b\vec{v} + c\vec{w}$.

In video game design, the lighting of a room depends on three spotlights whose beam directions are defined by the vectors $\vec{v_1} = (1, 2, k)$, $\vec{v_2} = (k, 0, 1)$ and $\vec{v_3} = (3, 2, 2)$. Determine, giving reasons, the values of the real parameter $k$ for which the three light beams are linearly dependent (lie in the same plane). What happens to the basis of space if $k$ takes one of those values?

Consider the vectors in space $\vec{u} = (1, 1, 0)$, $\vec{v} = (0, 1, 1)$ and $\vec{w} = (1, 0, 1)$.

1. a) Show that the set $B = \{ \vec{u}, \vec{v}, \vec{w} \}$ forms a basis for $\mathbb{R}^3$.
2. b) Find the coordinates of the standard basis vector $\vec{e_1} = (1, 0, 0)$ with respect to the basis $B$.

A laser security system uses three sensors located at points $A(1, 2, 3)$, $B(3, 0, 1)$ and $C(k, -2, -1)$. For the system to function correctly, the three sensors must be perfectly aligned.

1. a) Calculate the value of $k$ such that the sensors $A, B$ and $C$ are aligned.
2. b) Using the value of $k$ found, determine the coordinates of the midpoint of line segment $AC$.

Solar panels are being installed on a metallic structure. The anchor points of a panel are $P(1, 0, 1)$, $Q(2, 1, 3)$, $R(0, 2, 2)$ and $S(1, 3, a)$. For the panel not to suffer mechanical stress, the four points must be coplanar (lie in the same plane).

1. a) Determine the value of $a$ such that the points $P, Q, R$ and $S$ are coplanar.
2. b) For the value of $a$ obtained, are the vectors $\vec{PQ}$ and $\vec{PR}$ linearly independent? Justify your answer.

Given the points $A(2, -1, 4)$ and $B(8, 5, -2)$, find the coordinates of the points $C$ and $D$ that divide line segment $AB$ into three equal parts. Explain the vector relationship you used to solve it.

Study the rank of the following family of vectors depending on the values of the parameter $m$:

$\vec{u} = (1, m, 1)$, $\vec{v} = (m, 1, 1)$ and $\vec{w} = (1, 1, m)$.

Determine for which values of $m$ the set of vectors is linearly dependent and, if so, state whether the vectors are proportional or coplanar.

In a physics simulation software, a local reference frame is defined using the vectors $\vec{a} = (2, 1, -1)$, $\vec{b} = (1, 0, 1)$ and a third vector $\vec{c}$ such that the set is linearly independent.

If the vector $\vec{c}$ has the form $(0, 1, \lambda)$, determine the interval of values of $\lambda$ for which $\vec{c}$ cannot be expressed as a linear combination of $\vec{a}$ and $\vec{b}$. Reason whether this means that the set $\{ \vec{a}, \vec{b}, \vec{c} \}$ would form a basis for $\mathbb{R}^3$.

A surveillance drone takes off from a platform located at point $A(1, -2, 5)$ and travels in a straight line to reach a target at point $B(4, 3, -1)$. Calculate the components of the displacement vector $\vec{AB}$ and obtain its magnitude to find the distance travelled by the drone.

Given the vectors $\vec{u} = (2, -1, 3)$ and $\vec{v} = (0, 4, 1)$, perform the following analytical operations:

1. a) $\vec{w} = \vec{u} + \vec{v}$

1. b) $\vec{t} = 3\vec{u} - 2\vec{v}$

1. c) Calculate the value of the scalar $k$ if it is known that $k \cdot \vec{u} = (10, -5, 15)$.

In the design of a metallic structure, two beams are joined at their ends represented by the points $P(2, 5, -3)$ and $Q(8, -1, 7)$. To reinforce the structure, a support must be placed exactly at the midpoint of line segment $PQ$. Find the coordinates of this midpoint $M$.

Determine whether the vectors $\vec{a} = (1, -2, 4)$ and $\vec{b} = (-3, 6, -12)$ are linearly dependent. Justify your answer by checking if their coordinates are proportional and explain what geometric relationship exists between them (are they parallel?).

Express the vector $\vec{w} = (7, 11, 2)$ as a linear combination of the standard basis vectors of $\mathbb{R}^3$, that is, in terms of $\vec{i} = (1, 0, 0)$, $\vec{j} = (0, 1, 0)$ and $\vec{k} = (0, 0, 1)$.

Three radio stations are located at points $A(1, 1, 1)$, $B(3, 2, 0)$ and $C(5, 3, -1)$. Check whether the three stations are collinear (belong to the same straight line) by studying the proportionality of the vectors $\vec{AB}$ and $\vec{BC}$.

Calculate the value of the parameter $m$ so that the vector $\vec{u} = (m, 2, 4)$ is a linear combination of the vectors $\vec{v} = (1, 0, 2)$ and $\vec{w} = (0, 1, -1)$ through the expression $\vec{u} = 3\vec{v} + 2\vec{w}$.

Consider a parallelogram where three of its consecutive vertices are $A(1, 1, 0)$, $B(4, 2, 1)$ and $C(5, 5, 3)$. Using the equality between vectors (for example, $\vec{AB} = \vec{DC}$), calculate the coordinates of the fourth vertex $D$.

The points in space $A(1,\; -1,\; 2)$ and $B(3,\; 1,\; 0)$ are given. The goal is to determine the position of a third point $P(x,\; y,\; z)$ that simultaneously satisfies three conditions:

1. Points $A$, $B$, and $P$ are collinear (lie on the same line).

2. The distance between $A$ and $P$ is exactly three times the distance between $A$ and $B$ (that is, $|\vec{AP}| = 3\cdot |\vec{AB}|$).

3. The third coordinate of $P$ is strictly negative ($z < 0$).

1. a) Find the components of vector $\vec{AB}$ and calculate its exact magnitude.
2. b) Set up the vector relationship between $\vec{AP}$ and $\vec{AB}$, deducing the two possible displacement vectors $\vec{AP}$ that satisfy the distance condition.
3. c) Reasonably determine the coordinates of point $P$, ruling out the solution that does not satisfy the restriction on the $z$-coordinate.

The State Meteorological Agency (AEMET) tracks a weather balloon launched from the vicinity of Madrid. A Cartesian coordinate system is established with origin $O(0,\; 0,\; 0)$ at the launch station, where the $X$-axis points east, the $Y$-axis points north (both with distances in kilometers), and the $Z$-axis measures altitude (in hundreds of meters, so that $z = 10$ represents $1000\text{ m}$ of altitude).

During an initial ascent phase, the balloon reaches position $A(0,\; 0,\; 12)$. Next, a steady wind current moves it according to the displacement vector $\vec{u} = (4,\; 3,\; 8)$ to a position $B$.

1. a) Determine the coordinates of position $B$ and indicate the balloon's altitude in meters at that point.
2. b) A tracking team wishes to set up an observation point $M$ exactly halfway along the path between the launch point $O$ and position $B$. Find the coordinates of $M$ and calculate the direct straight-line distance from the origin $O$ to $M$, expressing the result in kilometers (take into account the unit conversion: $1\text{ unit of } z = 0,1\text{ km}$).
3. c) From position $B$, a gust pushes the balloon according to the vector $\vec{w} = (-2;\; -1,5;\; -4)$, moving it to a new point $C$. Prove vectorially that points $A$, $B$, and $C$ are collinear.

Given the vectors in space $\mathbb{V}^3$ defined by $\vec{u} = (k, 1, 1)$, $\vec{v} = (1, k, 1)$ and $\vec{w} = (1, 1, k)$:

• a) Determine, giving reasons, for which values of the real parameter $k$ the vectors $\{\vec{u}, \vec{v}, \vec{w}\}$ do not form a basis for $\mathbb{R}^3$.

• b) For the values of $k$ found in the previous part, study the linear dependence or independence of the vectors taken two at a time.

• c) In the case where $k = 0$, express the vector $\vec{a} = (2, 3, 5)$ as a linear combination of $\vec{u}$, $\vec{v}$ and $\vec{w}$.

Consider a basis $B = \{\vec{e_1}, \vec{e_2}, \vec{e_3}\}$ for an Euclidean vector space. Three new vectors are defined from the elements of the basis as:

$\vec{u} = \vec{e_1} + a\vec{e_2}$

$\vec{v} = \vec{e_2} + b\vec{e_3}$

$\vec{w} = \vec{e_3} + c\vec{e_1}$

Show that the vectors $\{\vec{u}, \vec{v}, \vec{w}\}$ are linearly dependent if and only if the product of the coefficients $abc = -1$. In the case where $a=1, b=1, c=-1$, obtain the linear dependence relation that exists between them.

A team of engineers is calibrating the navigation system of a satellite. Two different bases have been defined in the workspace: the control basis $B = \{\vec{u}, \vec{v}, \vec{w}\}$ and the manoeuvre basis $B' = \{\vec{a}, \vec{b}, \vec{c}\}$. We know that:

$\vec{a} = 2\vec{u} - \vec{v}$

$\vec{b} = \vec{u} + \vec{w}$

$\vec{c} = \vec{v} - \vec{w}$

A detected object has coordinates $(1, -2, 3)$ with respect to the manoeuvre basis $B'$. Calculate its coordinates with respect to the control basis $B$. Is it possible for a vector to have the same coordinates in both bases? Justify your answer.

In a tetrahedron with vertices $O(0,0,0)$, $A(2,2,0)$, $B(0,2,1)$, and $C(k,0,3)$, determine the value of $k$ such that the vector joining the origin $O$ to the centroid $G$ of the opposite face $ABC$ is perpendicular to the plane containing that face. Note: The centroid of a triangle is the arithmetic mean of its vertices.

Let $\vec{u} = (1, -1, 0)$ and $\vec{v} = (0, 1, -1)$ be vectors. Find all vectors $\vec{w}$ that simultaneously satisfy the following three conditions:

1. $\vec{w}$ is a linear combination of $\vec{u}$ and $\vec{v}$.

2. The magnitude of $\vec{w}$ is equal to $\sqrt{6}$.

3. $\vec{w}$ forms an angle of $60^\circ$ with the vector $\vec{u}$.

Study the linear dependence of the points $A(1, 1, 1)$, $B(0, 2, 2)$, $C(2, 0, 0)$ and $D(m, m, m)$.

• a) Determine for which values of $m$ the four points are coplanar.

• b) Is there any value of $m$ for which the vector $\vec{AB}$ is orthogonal to the vector $\vec{CD}$?

• c) Geometrically explain what happens to the points when $m=1$.

Consider the set of vectors $S = \{ \vec{v_1}, \vec{v_2}, \vec{v_3} \}$ forming an orthonormal basis. A new set of vectors $S' = \{ \vec{w_1}, \vec{w_2}, \vec{w_3} \}$ is constructed using the relations:

$\vec{w_1} = \vec{v_1} + \vec{v_2}$

$\vec{w_2} = \vec{v_1} - \vec{v_2}$

$\vec{w_3} = \vec{v_3}$

• a) Show that $S'$ is an orthogonal basis but not an orthonormal one.

• b) Find the volume of the parallelepiped defined by the vectors of $S'$.

• c) If a vector $\vec{x}$ has coordinates $(3, 1, 4)$ in the basis $S$, calculate its coordinates in the basis $S'$.

Two particles move through space following straight-line trajectories. At time $t=0$, the first particle is at point $P(1, 0, 0)$ and moves with a constant velocity given by the vector $\vec{v_1} = (1, 1, 1)$. The second particle starts from $Q(0, 2, 1)$ with velocity $\vec{v_2} = (2, m, 0)$.

Determine the value of $m$ such that the position vectors of both particles at any time $t > 0$ and the vector joining their initial positions $\vec{PQ}$ are always coplanar. Will the particles collide at any point for that value of $m$?

To determine whether three points in space $A(1,\; 2,\; -1)$, $B(3,\; -2,\; 3)$, and $C(6,\; -8,\; 9)$ are collinear (lie on the same line), two different resolution strategies are proposed:

• Method 1 (Metric): Calculate the Euclidean distances $d(A, B)$, $d(B, C)$, and $d(A, C)$ using the magnitude of their respective vectors and check whether the sum of the two smaller distances equals the largest one exactly.

• Method 2 (Vector): Form the vectors $\vec{AB}$ and $\vec{AC}$ and analyze whether they are linearly dependent by checking the proportionality of their components: $\frac{b_1-a_1}{c_1-a_1} = \frac{b_2-a_2}{c_2-a_2} = \frac{b_3-a_3}{c_3-a_3}$.

1. a) Apply Method 2 to decide with justification whether points $A$, $B$, and $C$ are collinear or not.
2. b) Apply Method 1 by calculating the magnitudes $|\vec{AB}|$ and $|\vec{AC}|$, simplifying the resulting radicals, and verify that the same conclusion is reached.
3. c) Critically compare both methods and write a reasoned justification regarding which one is more convenient. In your analysis, consider two aspects: the algebraic cost (presence of square roots versus direct ratios) and the ease of application in the case where the third coordinate of $C$ contains an unknown literal parameter $k$.

Consider the space vectors $\vec{u} = (1,\; -1,\; 2)$ and $\vec{v} = (0,\; 2,\; 1)$, as well as the vector dependent on a real parameter $k$: $\vec{w}_k = (2,\; k,\; 7)$.

1. a) Set up the system of three equations with two unknowns ($a, b$) that expresses the condition for $\vec{w}_k$ to be a linear combination of $\vec{u}$ and $\vec{v}$, that is, $\vec{w}_k = a\vec{u} + b\vec{v}$.
2. b) Solve the equations that are independent of $k$ to determine the only possible coefficients $a$ and $b$. From them, deduce whether there is any value of the parameter $k$ for which $\vec{w}_k$ is a linear combination of $\vec{u}$ and $\vec{v}$.
3. c) Now suppose that the vector $\vec{t} = (3,\; -3,\; 6)$ is defined. Without solving any system of equations, reason whether $\vec{t}$ is linearly dependent on $\vec{u}$ and describe the relative position of the lines containing both vectors in space.
4. d) Formally and reasonably justify why no pair of non-zero real scalars $\alpha, \beta \neq 0$ can satisfy $\alpha\vec{u} + \beta\vec{v} = (0,\; 0,\; 0)$, relating your argument to the linear independence of $\vec{u}$ and $\vec{v}$.