Exercises: Linear combinations of vectors. Bases

Given the vectors $\vec{u} = (2, -3)$ and $\vec{v} = (-4, 6)$, mathematically justify whether they can form a basis for the vector space $\mathbb{R}^2$.

Calculate the coordinates of the vector resulting from the linear combination $3\vec{a} - 2\vec{b}$, knowing that $\vec{a} = (1, 4)$ and $\vec{b} = (-2, 5)$.

Determine the value of the parameter $m$ such that the vectors $\vec{p} = (m, 4)$ and $\vec{q} = (3, -2)$ are linearly dependent.

Express the vector $\vec{c} = (5, -2)$ as a linear combination of the standard basis vectors $\vec{i} = (1, 0)$ and $\vec{j} = (0, 1)$.

Check whether the set of vectors $B = \{(-1, 2), (2, 1)\}$ forms an orthogonal basis for the plane.

Knowing that $\vec{w} = 2\vec{u} + k\vec{v}$, with $\vec{u} = (1, -1)$, $\vec{v} = (0, 3)$ and $\vec{w} = (2, 7)$, find the value of the scalar $k$.

Check whether the vectors $\vec{u} = (3, 0)$ and $\vec{v} = (0, -2)$ form an orthogonal basis for $\mathbb{R}^2$.

Reason whether three non-zero vectors in the plane, for example $\vec{u} = (1, 1)$, $\vec{v} = (-1, 2)$ and $\vec{w} = (3, 0)$, can be simultaneously linearly independent.

A student, Marcos, has to find out whether the vectors $\vec{u} = (2, -1)$ and $\vec{v} = (-6, 3)$ form a basis of the plane $\mathbb{R}^2$. On his exam, he writes the following:

“We calculate the dot product: $\vec{u} \cdot \vec{v} = 2 \cdot (-6) + (-1) \cdot 3 = -12 - 3 = -15$. Since the dot product is not zero, the vectors are not perpendicular and, therefore, they do not have the same direction; this means that they are linearly independent and form a basis of $\mathbb{R}^2$”.

Identify the conceptual mistake Marcos makes, explain rigorously why his deduction is false, and write the correct mathematical justification to determine whether or not $\vec{u}$ and $\vec{v}$ form a basis.

Consider the vectors $\vec{u} = (2, 1)$ and $\vec{v} = (-1, 2)$, which form a basis $B = \{\vec{u}, \vec{v}\}$ of $\mathbb{R}^2$, and the target vector $\vec{w} = (3, 4)$.

1. a) Without solving any system of equations, reasonably estimate the signs of the scalars $a$ and $b$ in the linear combination $\vec{w} = a\vec{u} + b\vec{v}$, and argue which of the two coefficients should have a greater absolute value by observing the quadrants towards which each vector points.
2. b) Set up the system of linear equations, calculate exactly the coordinates $(a, b)_B$ of $\vec{w}$ in that basis, and check whether the numerical result matches your previous estimate.

Two tugboats pull a disabled ship with forces $\vec{F}_1 = (300, 400)$ N and $\vec{F}_2 = (400, -300)$ N. A third tugboat joins by applying a force $\vec{F}_3 = (1000, 500)$ N. Express the force $\vec{F}_3$ as a linear combination of the vectors $\vec{F}_1$ and $\vec{F}_2$, finding the coordinates of $\vec{F}_3$ in the basis $B = \{\vec{F}_1, \vec{F}_2\}$.

In the physics engine of a video game, the movement of a character is defined in a basis $B = \{\vec{u}, \vec{v}\}$, where $\vec{u} = (2, 1)$ and $\vec{v} = (-1, 3)$. If the character must move to the point indicated by the vector $\vec{d} = (8, -3)$ relative to the standard basis, what will be the coordinates of the displacement vector in the basis $B$?

Given the parallelogram $ABCD$ in the figure, express the vectors corresponding to the diagonals $\vec{AC}$ and $\vec{BD}$ as a linear combination of the side vectors $\vec{u} = \vec{AB}$ and $\vec{v} = \vec{AD}$.

A logistics company uses two types of vans. The load capacity of type A vans is represented by the vector $\vec{a} = (3, 2)$, indicating 3 pallets of fruit and 2 of vegetables. Type B is represented by $\vec{b} = (1, 4)$. If a shopping centre places an exact order of 11 pallets of fruit and 14 of vegetables, represented by $\vec{p} = (11, 14)$, find the scalars of the linear combination $\vec{p} = x\vec{a} + y\vec{b}$ to determine how many trips each van needs to make.

Determine a vector $\vec{v}$ such that the set $B = \{\vec{u}, \vec{v}\}$ forms an orthonormal basis, given that $\vec{u} = \left(\frac{3}{5}, \frac{-4}{5}\right)$. Is the solution unique? Justify your answer.

An underwater drone moves in an environment with ocean currents. Its autonomous navigation system uses a basis formed by the vectors $\vec{c}_1 = (1, 2)$ and $\vec{c}_2 = (-2, 3)$. The drone records a displacement $\vec{w}$ whose coordinates in this basis are $(4, -1)_B$. Calculate the coordinates of the vector $\vec{w}$ in the standard basis.

Find the values of $a$ and $b$ so that the vectors $\vec{u} = (a, 3)$ and $\vec{v} = (-2, b)$ form an orthogonal basis, given also that the magnitude of the vector $\vec{u}$ is exactly $5$.

The attached plane shows the vectors $\vec{u}$ and $\vec{v}$. Determine the coordinates of the resultant vector $\vec{z} = 2\vec{u} - \vec{v}$ in the standard basis.

At the Salvamento Marítimo control centre in Tarifa (Cádiz), an operator monitors a rescue drone in the Strait. Due to local environmental conditions, the autopilot decomposes movements into two reference directions: that of the prevailing Levante wind $\vec{v}_L = (4, 1)$ and that of the surface sea current $\vec{v}_C = (-1, 3)$ (both velocities expressed in $\text{m/s}$ relative to standard orthonormal axes).

1. a) Analytically justify that $B = \{\vec{v}_L, \vec{v}_C\}$ forms a basis of $\mathbb{R}^2$ and check whether it is an orthogonal basis.
2. b) The drone must reach a resultant ground velocity of $\vec{v}_{req} = (11, 6)\text{ m/s}$. Model this situation by expressing $\vec{v}_{req}$ as a linear combination of $\vec{v}_L$ and $\vec{v}_C$ to obtain its coordinates in the basis $B$.
3. c) The drone's thrusters can only provide a thrust factor in the direction of the current between $-1$ and $2$ units. In light of the coordinates obtained, interpret whether the manoeuvre is physically feasible with this configuration.

We want to express the vector $\vec{w} = (2, 11)$ as a linear combination of the vectors of the basis $B = \{\vec{u} = (3, 4), \vec{v} = (-4, 3)\}$ to find its coordinates $(a, b)_B$.

Two approaches are proposed to solve it:

• Method 1 (General algebraic): Set up the system $a(3, 4) + b(-4, 3) = (2, 11)$ and solve it by substitution, elimination, or equating.

• Method 2 (Taking advantage of orthogonality): Verify that $\vec{u} \cdot \vec{v} = 0$ and apply the fact that, in an orthogonal basis, taking the dot product on both sides of $\vec{w} = a\vec{u} + b\vec{v}$ with each basis vector directly yields $a = \frac{\vec{w} \cdot \vec{u}}{|\vec{u}|^2}$ and $b = \frac{\vec{w} \cdot \vec{v}}{|\vec{v}|^2}$.

1. a) Carry out the full calculations using both methods and verify that they lead to the same result.
2. b) Compare both procedures: justify which is more advantageous in terms of efficiency and lower likelihood of operational error, and explain why Method 2 would fail if the basis were not orthogonal.

Prove analytically that if $B = \{\vec{u}, \vec{v}\}$ is a basis of vectors in the plane, then the set $B' = \{\vec{u} + \vec{v}, \vec{u} - \vec{v}\}$ also forms a basis. If a vector $\vec{w}$ has coordinates $(3, 5)$ in the basis $B$, find its coordinates in the new basis $B'$.

To guarantee the stability of a node in a planar lattice structure, the direction vectors of two concurrent bars must be linearly independent. The design vectors are $\vec{d}_1 = (k+1, 3)$ and $\vec{d}_2 = (2, k)$. Determine for which values of the parameter $k$ the node is unstable (the vectors do not form a basis). Subsequently, assuming $k=0$, express the external load vector $\vec{P} = (4, -6)$ as a linear combination of $\vec{d}_1$ and $\vec{d}_2$.

In the regular hexagon $ABCDEF$ with centre $O$ shown in the figure, the basis vectors $\vec{u} = \vec{AB}$ and $\vec{v} = \vec{AF}$ are defined. Express the vectors $\vec{AC}$, $\vec{AD}$, $\vec{AE}$ and $\vec{BC}$ as a linear combination of the basis $B = \{\vec{u}, \vec{v}\}$.

In the study of a two-dimensional crystal lattice, two bases are defined for the plane: the primitive basis $B_1 = \{\vec{a}_1, \vec{a}_2\}$ and a secondary basis $B_2 = \{\vec{b}_1, \vec{b}_2\}$. Knowing that $\vec{b}_1 = 2\vec{a}_1 - \vec{a}_2$ and $\vec{b}_2 = \vec{a}_1 + 3\vec{a}_2$, find the coordinates in the basis $B_1$ of a point defect whose position vector in the basis $B_2$ is $\vec{d} = (-1, 2)_{B_2}$.

Let $\vec{u}$ and $\vec{v}$ be two non-zero vectors such that the angle between them is $60^\circ$, with $|\vec{u}| = 2$ and $|\vec{v}| = 3$. If the vector $\vec{w} = m\vec{u} + n\vec{v}$ is defined, calculate the relation that must exist between the parameters $m$ and $n$ so that the vector $\vec{w}$ is perpendicular to the vector $\vec{u}$.

The arm of an industrial robot moves in a plane defined by a basis $B = \{\vec{e}_1, \vec{e}_2\}$ which is orthogonal but not orthonormal, where $|\vec{e}_1| = 2$ and $|\vec{e}_2| = 4$. A path command instructs it to move according to the vector $\vec{v} = 3\vec{e}_1 - \frac{1}{2}\vec{e}_2$. Calculate the exact magnitude of the displacement vector $\vec{v}$ and the cosine of the angle it forms with the basis vector $\vec{e}_1$.

Find all vectors $\vec{x} = (x_1, x_2)$ that, when expressed as a linear combination of the basis $B = \{(1, -1), (2, 1)\}$, have coordinates $(a, b)_B$ such that $a = b$ and, furthermore, satisfy that their magnitude in the standard basis is exactly $\sqrt{45}$.

Given the points $A(1, 2)$, $B(4, -1)$ and $C(x, y)$, determine the coordinates of the point $C$ knowing that the vectors $\vec{u} = \vec{AB}$ and $\vec{v} = \vec{AC}$ form a basis of the plane in which the vector $\vec{w} = (0, -6)$ has coordinates $(2, -1)_B$.

A technical designer needs to set up an oblique coordinate system $B = \{\vec{u}, \vec{v}\}$ in the working plane. The following requirements are known:

1. The first basis vector has the form $\vec{u} = (1, k)$ and the second vector has the form $\vec{v} = (m, 1)$, with $k, m \in \mathbb{R}$.

2. The vectors must be perpendicular to each other, so that the basis $B$ is orthogonal.

3. An anchor point located at the canonical position $\vec{w} = (5, 5)$ must have coordinates in the basis $B$ of exactly $(a, b)_B = (3, -1)$.

1. a) Set up and solve the system of equations that allows you to determine the numerical values of $k$ and $m$.
2. b) Determine whether the obtained basis $B = \{\vec{u}, \vec{v}\}$ is orthonormal. If not, construct an orthonormal basis $B' = \{\vec{u}', \vec{v}'\}$ from it while keeping the same directions and senses, and calculate the new coordinates of $\vec{w}$ in $B'$.
3. c) Prove that the magnitude of vector $\vec{w}$ is the same whether calculated from its canonical components or from the standard Euclidean formula applied to its coordinates in the orthonormal basis $B'$, and briefly reflect on why this does not happen with its coordinates in basis $B$.

At a 061 operations base in the Aragonese Pyrenees, the air navigation team studies the trajectories of a rescue helicopter. Two usual valley winds are described by the vectors $\vec{v}_1 = (2, 1)$ and $\vec{v}_2 = (k, 3)$, measured in knots.

1. a) Find the value of the real parameter $k$ for which the vectors $\vec{v}_1$ and $\vec{v}_2$ are linearly dependent. Explain what geometric and operational limitation it would mean for the helicopter's navigation if $k$ were to take that critical value.
2. b) For $k = -1$, prove that $B = \{\vec{v}_1, \vec{v}_2\}$ is a basis of $\mathbb{R}^2$ and find the coordinates in that basis of the manoeuvre vector $\vec{w} = (5, 6)$.
3. c) A co-pilot makes the following general hypothesis: “If the same non-zero gust wind $\vec{r} = (1, 2)$ is added to both vectors of any basis $B = \{\vec{v}_1, \vec{v}_2\}$ of $\mathbb{R}^2$, the resulting new set $B' = \{\vec{v}_1 + \vec{r}, \vec{v}_2 + \vec{r}\}$ will always continue to be a basis of $\mathbb{R}^2$”. Prove whether the co-pilot's statement is true or false by algebraically analysing the linear dependence of $B'$ as a function of $k$ and providing an explicit counterexample, if applicable.