Exercises: Vectors and vector spaces

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 center 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 $D$.

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 $m$?