The Dot Product

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

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

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

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

A flat solar panel is oriented so that its normal vector (perpendicular to the surface) is $\vec{n} = (1, \sqrt{3})$. The sun's rays strike the panel following the direction of the vector $\vec{s} = (2, -1)$. Calculate the angle of incidence of the solar rays, defined as the angle formed by the normal vector $\vec{n}$ with the inverted solar ray vector $-\vec{s}$.

Calculate the value (or values) of the parameter $k$ so that the angle formed by the vectors $\vec{u} = (k, 3)$ and $\vec{v} = (1, -2)$ is exactly $45^\circ$.

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

Find the coordinates of a vector $\vec{w}$ that is perpendicular to the vector $\vec{v} = (-5, 12)$ and has the same modulus.

Determine the coordinates of a vector $\vec{v}$ that is orthogonal to the vector $\vec{u} = (3, -4)$ and whose modulus is exactly $10$. Bear in mind that there are two possible solutions; indicate both of them.

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

Calculate the modulus of the vector $\vec{w} = 3\vec{u} - 2\vec{v}$ knowing the components of the vectors $\vec{u} = (1, -1)$ and $\vec{v} = (0, 4)$.

Knowing that the moduli of two vectors are $|\vec{a}| = 3$ and $|\vec{b}| = 5$, and that their scalar product is $\vec{a} \cdot \vec{b} = -2$, calculate the exact value of the expression $(2\vec{a} + \vec{b}) \cdot (\vec{a} - 3\vec{b})$.

Prove, using exclusively 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 modulus ($|\vec{u}| = |\vec{v}|$). Express the diagonals in terms of $\vec{u}$ and $\vec{v}$.

Prove the "Parallelogram Law" analytically using the properties of the scalar product. That is, prove that for any two vectors $\vec{u}$ and $\vec{v}$, the following identity holds:

$|\vec{u} + \vec{v}|^2 + |\vec{u} - \vec{v}|^2 = 2|\vec{u}|^2 + 2|\vec{v}|^2$

Explain each algebraic step based on the definition of the squared modulus as the scalar product of a vector with itself.

It is known that $\vec{u}$ and $\vec{v}$ are two unit vectors. If the vectors constructed as $\vec{a} = \vec{u} + 2\vec{v}$ and $\vec{b} = 5\vec{u} - 4\vec{v}$ turn out to be orthogonal to each other, determine the exact angle formed by the original vectors $\vec{u}$ and $\vec{v}$.

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 rectilinear displacement given by the vector $\vec{d} = (12, 5)$ in feet. Knowing that mechanical work is defined as $W = \vec{F} \cdot \vec{d}$, calculate the work done by the crane and the angle formed by the applied force and the displacement.

In the development of a physics engine for a video game, a character moves following 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.

In computer graphics, the diffuse light intensity on a surface is calculated using Lambert's law: $I = I_0 \max(0, \vec{n} \cdot \vec{l})$, where $\vec{n}$ is the unit normal vector to the surface and $\vec{l}$ is the unit vector pointing toward the light source. If a surface has a normal vector with the same direction and sense as $\vec{v} = (-\sqrt{3}, 1)$ and the light source lies in the direction and sense of the vector $\vec{w} = (1, 1)$, calculate the exact light intensity $I$ as a fraction of $I_0$.

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

Consider the fixed points $A(0, 3)$ and $B(5, 2)$. Find all the points $C$ located on the x-axis such that the triangle $ABC$ is a right triangle with the right angle located at the vertex $C$.

In a triangle whose vertices are the points $P(-3, -1)$, $Q(5, -1)$ and $R(1, 6)$, use the concept of the scalar product and orthogonality of vectors to determine the exact coordinates of its orthocenter (the point where the three altitudes of the triangle intersect).

Find the value of $x$ so that the vector $\vec{u} = (x, 2)$ forms an angle of $30^\circ$ with the positive x-axis. Use the scalar product with the unit vector $\vec{i} = (1, 0)$ to set up the equation.

Find the coordinates of a vector $\vec{w} = (x, y)$ knowing that its scalar product with the vector $\vec{u} = (2, -1)$ is equal to $4$, and that its scalar product with the vector $\vec{v} = (1, 3)$ is equal to $-5$.

Given the vectors dependent on the parameter $t$, $\vec{a} = (t, 2)$ and $\vec{b} = (t-1, t)$, determine the value of $t$ that makes the scalar product $\vec{a} \cdot \vec{b}$ reach its minimum value. Also calculate the angle formed by the vectors $\vec{a}$ and $\vec{b}$ for that specific value of $t$.