Exercises: Scalar product

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

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 $\vec{a} \cdot \vec{b}$.

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

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

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

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 $|\vec{v}| = 3$.

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

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

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 meters. 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.

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 $135^\circ$.

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.

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.

Given that the magnitudes 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 $(2\vec{a} + \vec{b}) \cdot (\vec{a} - 3\vec{b})$.

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

A flat solar panel is oriented such that its normal vector (perpendicular to the surface) is $\vec{n} = (1, \sqrt{3})$. Sun 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 between the normal vector $\vec{n}$ and the inverted solar ray vector $-\vec{s}$.

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.

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 $\vec{v}$.

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 between the vectors $\vec{a}$ and $\vec{b}$ for that specific value of $t$.

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

In computer graphics, the intensity of diffuse light 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 towards 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 $I_0$.

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 between the original vectors $\vec{u}$ and $\vec{v}$.

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

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 magnitude as the scalar product of a vector with itself.