Linear combinations of vectors. Bases

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

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 $k$.

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

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

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.

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

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 $B$?

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

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

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.

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 $B = \{\vec{F}_1, \vec{F}_2\}$.

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 $B'$.