Scalar product

Given the points $A(1, 2)$, $B(4, 3)$ and $C(0, 5)$ in the plane, calculate the value of the scalar product $\vec{AB} \cdot \vec{AC}$.

Knowing that the magnitudes of two vectors are $|\vec{a}| = 4$ and $|\vec{b}| = 6$, and that the angle between them is $60^\circ$, calculate the exact value of their scalar product $\vec{a} \cdot \vec{b}$.

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

Calculate the value (or values) of the parameter $k$ such that the angle formed by the vectors $\vec{u} = (k, 3)$ and $\vec{v} = (1, -2)$ is exactly $135^\circ$.

Given the vectors $\vec{u} = (3, -2)$ and $\vec{v} = (4, 5)$, calculate their scalar product and determine whether they are orthogonal.

Find the value of the parameter $m$ so that the vectors $\vec{p} = (m, -3)$ and $\vec{q} = (6, 2)$ are perpendicular.

Check, using only the scalar product, whether the triangle with vertices $A(-2, 1)$, $B(4, 3)$ and $C(1, 12)$ is right-angled, acute-angled or obtuse-angled. If it is right-angled, identify at which vertex the right angle is located.

On a construction site, a crane drags a concrete block by applying a constant force represented by the vector $\vec{F} = (450, 600)$ expressed in newtons. The block undergoes a straight-line displacement given by the vector $\vec{d} = (12, 5)$ in metres. Knowing that mechanical work is defined as $W = \vec{F} \cdot \vec{d}$, calculate the work done by the crane and the angle between the applied force and the displacement.

In the development of a physics engine for a video game, a character moves in the direction and sense of the vector $\vec{d} = (8, 4)$. Gravity in this virtual environment is defined by the vector $\vec{g} = (0; -9,8)$. Calculate the orthogonal projection of the gravity vector onto the direction of the character's movement.

Using the properties of the scalar product, expand and calculate the value of the expression $(\vec{u} - \vec{v}) \cdot (\vec{u} + \vec{v})$ given that $|\vec{u}| = 5$ and $|\vec{v}| = 3$.

Prove, using only the algebraic properties of the scalar product, that the diagonals of any rhombus are perpendicular to each other.

Note: You can model the sides of the rhombus using two vectors $\vec{u}$ and $\vec{v}$ with the same origin and the same magnitude ($|\vec{u}| = |\vec{v}|$). Express the diagonals in terms of $\vec{u}$ and $\vec{v}$.