Vectors in the plane

Given the points $A(2, -1)$ and $B(5, 3)$, calculate the components of vector $\vec{AB}$ and its magnitude.

Look at the vectors $\vec{u}$ and $\vec{v}$ represented on the grid. Write down their components and analytically calculate the coordinates of the sum vector $\vec{s} = \vec{u} + \vec{v}$.

Find the possible values of $x$ so that the vector $\vec{v} = (x, -8)$ has a magnitude exactly equal to $10$.

A delivery drone flies with its own velocity represented by the vector $\vec{v}_d = (12, 5)$ km/h. At that instant, the wind blows with a velocity $\vec{v}_v = (-3, 2)$ km/h. Calculate the actual velocity vector of the drone and its speed (the magnitude of the resultant vector).

State whether the vectors $\vec{a} = (6, -9)$ and $\vec{b} = (-2, 3)$ are parallel (have the same direction). Justify your answer mathematically.

In the design of a bridge, three pillars must be perfectly collinear. On the architect's plan, the bases of the pillars are situated at the coordinates $P(-2, 1)$, $Q(2, 3)$ and $R(8, k)$. Determine the value of $k$ so that they are correctly aligned.

Calculate the scalar product of the vectors $\vec{p} = (4, 5)$ and $\vec{q} = (-2, 3)$. From the result, state whether they are perpendicular.

Consider the vectors depending on a parameter $\vec{u} = (m, 2)$ and $\vec{v} = (3, m-1)$. Find the values of the parameter $m$ in two different geometric cases:

1. a) So that $\vec{u}$ and $\vec{v}$ are perpendicular.
2. b) So that $\vec{u}$ and $\vec{v}$ are parallel.

Find out whether the triangle formed by the telecommunication antennas located at coordinates $A(1, 2)$, $B(4, 6)$ and $C(9, -1)$ is a right-angled triangle. Use the scalar product of the vectors that form its sides to justify your answer.