Linear Dependence and Independence. Bases of R³

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.

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.

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.

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.

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.

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.

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 $\vec{c} = (1, 1, m)$.

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:

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.

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 $k$?

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 $B' = \{\vec{u}, \vec{v}, \vec{w}\}$?

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 $B = \{ \vec{a}, \vec{b}, \vec{c} \}$.

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).

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.

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.