Exercises: Dependence and independence. Bases

Two surveillance drones move through space along straight trajectories defined by the vectors $\vec{u} = (2, -4, 6)$ and $\vec{v} = (-1, 2, -3)$. Determine whether the drones move in parallel directions by analysing the linear dependence of their direction vectors.

In a video game graphics engine, the position of an object is defined by a linear combination of two basic vectors: $\vec{a} = (1, 0, 2)$ and $\vec{b} = (0, -1, 3)$. Calculate the coordinates of the resulting vector $\vec{w}$ if the following operation is applied:

Given the vectors $\vec{u} = (1, 2, 3)$, $\vec{v} = (2, 0, 1)$ and $\vec{w} = (3, 2, 4)$, prove without performing complex calculations (by observing the relationship between their components) that the set $\{\vec{u}, \vec{v}, \vec{w}\}$ is linearly dependent.

A laser sensor system requires two beams of light to be parallel. The first beam follows the direction of vector $\vec{d_1} = (3, k, -1)$ and the second follows vector $\vec{d_2} = (6, -4, -2)$. Find the value of parameter $k$ so that the vectors are linearly dependent and, therefore, the beams are parallel.

Check whether the vectors $\vec{e_1} = (1, 0, 0)$, $\vec{e_2} = (1, 1, 0)$ and $\vec{e_3} = (1, 1, 1)$ can form a basis of $\mathbb{R}^3$. To do this, justify whether they are linearly independent using the determinant of the matrix they form.

Consider the points in space $A(1, 2, -1)$, $B(2, 3, 1)$ and $C(3, 4, 3)$.

1. a) Find the vectors $\vec{AB}$ and $\vec{AC}$.
2. b) From their components, decide whether points $A, B$ and $C$ are collinear.

If a vector $\vec{v}$ has coordinates $(3, -2, 5)$ with respect to the standard basis $\{\vec{i}, \vec{j}, \vec{k}\}$, express it as a linear combination of these unit vectors.

State with reasons whether the following statement is true or false: "In three-dimensional space $\mathbb{R}^3$, any set of four non-zero vectors is always linearly dependent." Give a simple example to support your answer.

Given the vectors $\vec{u} = (1, 2, -1)$ and $\vec{v} = (0, 1, 2)$, determine the value of the real parameter $k$ such that vector $\vec{w} = (2, k, 0)$ is a linear combination of $\vec{u}$ and $\vec{v}$. What geometrical relationship do the three vectors have for that value of $k$?

Study the linear dependence or independence of the set of vectors $S = \{\vec{a}, \vec{b}, \vec{c}\}$ according to the values of the parameter $m \in \mathbb{R}$, where $\vec{a} = (m, 1, 1)$, $\vec{b} = (1, m, 1)$, and $\vec{c} = (1, 1, m)$.

Give a reasoned proof that if vectors $\{\vec{u}, \vec{v}, \vec{w}\}$ are linearly independent, then vectors $\{\vec{u} + \vec{v}, \vec{v} + \vec{w}, \vec{w} + \vec{u}\}$ are also linearly independent and, therefore, form a basis of $\mathbb{R}^3$.

A surveillance drone moves along straight trajectories defined by vectors. If the direction vectors of three drones are $\vec{d_1} = (1, 3, a)$, $\vec{d_2} = (4, 1, 2)$ and $\vec{d_3} = (a, -1, 0)$, calculate the values of $a$ for which the trajectories of the three drones would be contained in planes parallel to each other (that is, the vectors are coplanar).

Consider the points in space $A(1, 0, 0)$, $B(0, 2, 1)$, and $C(2, 0, m)$.

1. a) Find the vector $\vec{AB}$ and the vector $\vec{AC}$.
2. b) Determine the value of $m$ such that the vector $\vec{v} = (3, -3, -3)$ is linearly dependent on $\vec{AB}$ and $\vec{AC}$.
3. c) For the value of $m$ found, write $\vec{v}$ as a linear combination of $\vec{AB}$ and $\vec{AC}$.

Given the vectors of the standard basis $\{\vec{i}, \vec{j}, \vec{k}\}$, the following vectors are defined:

Justify whether the set $\{\vec{u}, \vec{v}, \vec{w}\}$ is a basis of $\mathbb{R}^3$. If a vector $\vec{x}$ has coordinates $(1, 2, 3)$ in the standard basis, what will its coordinates be in the basis $B' = \{\vec{u}, \vec{v}, \vec{w}\}$?

Determine whether the points $P(1, 2, -1)$, $Q(3, 1, 1)$, $R(0, 5, -2)$ and $S(k, 0, 4)$ are coplanar depending on the value of $k$. To carry out this analysis, use the linear dependence of the vectors originating at $P$ with endpoints at the other points.

A team of engineers is designing the navigation system for a satellite. In order for the satellite to orient itself in any direction in three-dimensional space, it needs three thrusters whose thrust directions are linearly independent. Three thrusters defined by the vectors $\vec{u} = (1, m, 1)$, $\vec{v} = (m, 1, 1)$ and $\vec{w} = (1, 1, m)$ have been installed.

Determine, giving reasons, for which values of the real parameter $m$ the propulsion system would not be able to manoeuvre in all of space (that is, when the vectors are linearly dependent). In the case where $m=1$, explain geometrically what happens to the thrust directions and whether they form a basis for $\mathbb{R}^3$.

Given the vectors in space $\vec{a} = (1, 2, -1)$, $\vec{b} = (0, 1, 2)$, and $\vec{c} = (k, 1, -3)$, answer the following:

1. a) Find the value of $k$ so that the vector $\vec{v} = (3, 4, -7)$ can be expressed as a linear combination of $\vec{a}$, $\vec{b}$, and $\vec{c}$ such that the three vectors are linearly dependent.

1. b) If $k = 2$, do they form a basis of $\mathbb{R}^3$? If so, calculate the coordinates of the vector $\vec{w} = (1, 1, 1)$ with respect to this basis $B = \{ \vec{a}, \vec{b}, \vec{c} \}$.

In open-world video game development, the position of a character is defined using a non-standard basis $B' = \{ \vec{u_1}, \vec{u_2}, \vec{u_3} \}$. We know that these vectors are related to the standard basis $B = \{ \vec{i}, \vec{j}, \vec{k} \}$ as follows:

$\vec{u_1} = \vec{i} + \vec{j}$

$\vec{u_2} = \vec{j} + \vec{k}$

$\vec{u_3} = \vec{i} + \vec{k}$

1. a) Show that $B'$ is a basis for $\mathbb{R}^3$.

1. b) If an object in the game has coordinates $(2, -4, 6)$ in the standard basis, determine its coordinates in the basis $B'$.

Study the linear dependence or independence of the following vectors according to the values of the parameters $a$ and $b$:

$\vec{u} = (a, 1, 0)$

$\vec{v} = (1, a, 1)$

$\vec{w} = (0, 1, b)$

Determine for which relationship between $a$ and $b$ the vectors are coplanar. Is there any value of $a$ for which the vectors are linearly dependent regardless of the value of $b$? Justify your answer.

Let $\{\vec{u}, \vec{v}, \vec{w}\}$ be a basis of $\mathbb{R}^3$. Consider the vectors:

$\vec{x} = \vec{u} + \vec{v}$

$\vec{y} = \vec{v} + \vec{w}$

$\vec{z} = \vec{w} + \lambda\vec{u}$

1. a) For which values of $\lambda$ does the set $\{\vec{x}, \vec{y}, \vec{z}\}$ not form a basis of $\mathbb{R}^3$?

1. b) For $\lambda = 0$, express the vector $\vec{w}$ as a linear combination of $\vec{x}$, $\vec{y}$ and $\vec{u}$.

An architect designs a tensile structure using three cables that converge at an anchor point $P$. The directions of the cables are given by the vectors $\vec{v_1} = (1, 1, a)$, $\vec{v_2} = (a, 0, 1)$ and $\vec{v_3} = (2, 1, 1)$.

1. a) If the anchor is to be stable, the cables cannot all lie in the same plane. Find the values of $a$ that would cause the structure to collapse because the vectors are linearly dependent.

1. b) For $a = 2$, a total force of $\vec{F} = (5, 2, 4)$ is applied at point $P$. Determine whether this force can be uniquely decomposed as a sum of tensions in the directions of the cables and, if so, find this decomposition.

Consider the points $A(1, 0, 1)$, $B(0, 2, 1)$ and $C(2, 1, 0)$.

1. a) Find a point $D$ such that vector $\vec{AD}$ is linearly dependent on vector $\vec{BC}$ and has a magnitude of 6. Is the solution unique?

1. b) Determine whether point $E(5, -1, 4)$ belongs to the plane defined by vectors $\vec{AB}$ and $\vec{AC}$. Justify your answer using the concepts of linear dependence and basis.

Prove that if three vectors $\vec{u}, \vec{v}, \vec{w}$ are linearly independent, then vectors $\vec{u} - \vec{v}$, $\vec{v} - \vec{w}$ and $\vec{w} - \vec{u}$ are always linearly dependent.

Give a geometrical interpretation of this result, considering the vectors as the sides of a closed figure in space.