Dot product exercises

Calculate the scalar product of the vectors $\vec{u} = (3, -2)$ and $\vec{v} = (-1, 5)$.

Find the magnitude of the vector $\vec{w} = (-5, 12)$.

Determine, justifying your answer by means of a mathematical calculation, whether the vectors $\vec{p} = (4, 6)$ and $\vec{q} = (-3, 2)$ are orthogonal.

Calculate the value of the parameter $k$ so that the vectors $\vec{u} = (k, 3)$ and $\vec{v} = (2, -4)$ are perpendicular.

Find the coordinates of a vector that is perpendicular to $\vec{v} = (-7, 2)$ and has exactly the same magnitude.

In the physics engine of a video game, a character located at coordinate $P(10, 15)$ must fire a projectile in the direction of vector $\vec{d} = (3, -4)$ at a constant speed of $20$ distance units per frame. Determine the exact velocity vector $\vec{v}$ of the projectile, knowing that it must have the same direction and sense as $\vec{d}$ but with magnitude $20$.

Find the coordinates of a vector $\vec{w} = (x, y)$ knowing that its magnitude is $\sqrt{13}$ and that its scalar product with the vector $\vec{v} = (2, 3)$ gives $13$.

In physics, the work $W$ done by a constant force $\vec{F}$ in moving an object along a displacement vector $\vec{d}$ is calculated using the scalar product $W = \vec{F} \cdot \vec{d}$. If a crane applies a force of $\vec{F} = (150, -40)$ newtons to move a container from point $A(2, 5)$ to point $B(12, 5)$ (coordinates expressed in metres), calculate the total work done.

Calculate the orthogonal projection vector of the vector $\vec{u} = (4, -3)$ onto the vector $\vec{v} = (1, 2)$.

Calculate the angle between the vectors $\vec{a} = (1, \sqrt{3})$ and $\vec{b} = (-\sqrt{3}, 1)$.

To stabilise the mast of a telecommunications antenna, two cables are tensioned from the same point on the mast and anchored to the ground. The vectors representing the direction of the cables from the top anchor to the ground are $\vec{T_1} = (-4, -5)$ and $\vec{T_2} = (6, -3)$. Calculate the angle between the two cables at the top anchor point.

Calculate the distance between the points $A(2, -1)$ and $B(5, 3)$ by first determining the components of the vector $\vec{AB}$.

Check, using only the scalar product, whether the triangle defined by the vertices $A(1, 2)$, $B(4, 3)$ and $C(3, -1)$ is a right-angled triangle. If so, state at which vertex the right angle is located.