Exercises: Scalar product exercises

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

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

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.

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

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

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

Lucía wants to find a vector perpendicular to $\vec{u} = (4, -3)$ that has exactly the same magnitude. In her exam, she writes: "To make two vectors perpendicular, it is enough to change the signs of their coordinates, so the vector we are looking for is $\vec{v} = (-4, 3)$".\na) Explain Lucía's conceptual error in detail by calculating the dot product $\vec{u} \cdot \vec{v}$ and state what geometric relationship $\vec{u}$ and $\vec{v}$ actually have.\nb) Write the components of the two vectors that are indeed perpendicular to $\vec{u}$ and have the same magnitude, proving using the dot product why the component-swap rule $(-u_2, u_1)$ guarantees perpendicularity.

Determine the components of a vector $\vec{w}$ that simultaneously satisfies the following three conditions:\n1) It is perpendicular to the vector $\vec{u} = (3, -4)$.\n2) Its first component (x-coordinate) is strictly positive.\n3) Its magnitude is exactly $15$.\nTo solve it, first obtain a unit vector perpendicular to $\vec{u}$ and reason how to transform it so that it reaches the required magnitude without changing its direction or sense.

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 $\vec{v} = (1, 2)$.

Determine the possible values of the parameter $m$ so that the vectors $\vec{u} = (m, 1)$ and $\vec{v} = (2, m)$ form an angle of $30^\circ$.

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 $13$.

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.

Given the vectors $\vec{a} = (3, -1)$ and $\vec{b} = (1, 3)$, express the vector $\vec{c} = (7, 1)$ as a linear combination of $\vec{a}$ and $\vec{b}$. Once solved, calculate the scalar product of $\vec{a}$ and $\vec{b}$ and interpret the result geometrically.

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.

To check whether the triangle with vertices $A(1, 2)$, $B(5, 5)$, and $C(-2, 6)$ is a right-angled triangle, two classmates propose different strategies:\n- Irene proposes calculating the vectors of the sides sharing each vertex and checking whether the dot product of any pair of them is equal to $0$.\n- David proposes calculating the length of the three sides using distances between points and checking whether the resulting measurements satisfy the Pythagorean theorem ($a^2 + b^2 = c^2$).\na) Apply Irene's method by calculating the corresponding vectors and determine at which vertex the right angle is located, if it exists.\nb) Analyze and justify which of the two methods is more advantageous in coordinate geometry, considering the number of operations required and the handling of square roots.

At the Higueruela wind farm (Albacete), technicians monitor the movement of a maintenance trolley along a straight guide rail from point $A(1, 4)$ to point $B(7, 12)$, with coordinates expressed in meters. During this journey, the wind exerts a constant force on the trolley recorded by the sensors as $\vec{F} = (80, -30)$ Newtons.\na) Determine the displacement vector $\vec{d} = \vec{AB}$ and calculate the mechanical work $W = \vec{F} \cdot \vec{d}$ exerted by the wind.\nb) Interpret the sign of the work obtained: does the wind facilitate or hinder the trolley's progress?\nc) Calculate the angle $\alpha$ formed by the force vector $\vec{F}$ and the displacement vector $\vec{d}$ (rounding to the nearest tenth of a degree) and justify whether this angle is consistent with your interpretation of the sign of the work.

For the vectors $\vec{u}$ and $\vec{v}$, we know that $|\vec{u}| = 3$, $|\vec{v}| = 5$ and that they form an angle of $120^\circ$. Applying the properties of the scalar product, calculate the exact value of the expression $(2\vec{u} - \vec{v}) \cdot (\vec{u} + 3\vec{v})$.

An industrial robotic arm moves in a two-dimensional plane. Its gripper is initially at the point $P(2, 5)$ and must move in a straight line to a point $Q$ located on a conveyor belt whose path is defined by the line $x - 2y + 4 = 0$. If the displacement vector $\vec{PQ}$ must be orthogonal to the direction of the conveyor belt, determine the exact coordinates of the point $Q$.

Prove vectorially, using the properties of the scalar product, that the diagonals of any rhombus intersect perpendicularly. To do so, consider a rhombus generated by two free vectors $\vec{u}$ and $\vec{v}$ starting from the same vertex and satisfying $|\vec{u}| = |\vec{v}|$.

Find the coordinates of all vectors $\vec{a} = (x, y)$ that form an exact angle of $45^\circ$ with the vector $\vec{b} = (2, -2)$ and whose magnitude is exactly half the magnitude of $\vec{b}$.

In the installation of a solar panel, the normal vector (perpendicular to the surface of the panel) is given by $\vec{n} = (k, \sqrt{3})$. Sun rays strike the panel along the vector $\vec{s} = (1, -1)$. Calculate the positive value of the parameter $k$ so that the angle of incidence (the angle formed between the normal vector $\vec{n}$ and the vector opposite to the sun rays $-\vec{s}$) is $60^\circ$.

Given the points $A(-2, 1)$ and $B(4, 5)$, find the coordinates of a point $C$ situated on the x-axis such that the triangle $ABC$ is right-angled with the right angle at vertex $C$. How many possible solutions are there for point $C$?

Knowing that $\vec{u}$ and $\vec{v}$ are two unit vectors, determine the angle they must form with each other so that the sum vector $\vec{u} + \vec{v}$ has a magnitude equal to $\sqrt{3}$.

Calculate the exact area of the triangle determined by the vertices $A(1, 1)$, $B(5, 2)$ and $C(3, 6)$. To do so, use the orthogonal projection of the vector $\vec{AC}$ onto the vector $\vec{AB}$ to determine the height of the triangle, and the magnitude of $\vec{AB}$ as the base.

Consider the fixed vectors in the plane $\vec{u} = (1, 5)$ and $\vec{v} = (2, 1)$. For each value of the real parameter $t$, the combined vector $\vec{w}(t) = \vec{u} + t\vec{v}$ is defined. The goal is to find the exact value of $t$ for which $\vec{w}(t)$ is perpendicular to $\vec{v}$.\na) Before calculating, make a reasoned qualitative estimate: sketch the relative positions of $\vec{u}$ and $\vec{v}$ in the first quadrant. Since they form an acute angle, should the parameter $t$ be positive or negative for the linear combination to achieve a direction orthogonal to $\vec{v}$? Provide a reasoned estimate of which two consecutive integers $t$ must lie between.\nb) State the algebraic condition using the dot product $\vec{w}(t) \cdot \vec{v} = 0$ and find the exact value of $t$.\nc) Check whether the numerical result confirms your previous estimate and explain geometrically why this vector $\vec{w}(t)$ represents the segment of minimum length connecting the tip of $\vec{u}$ to the line containing $\vec{v}$.

In the airspace near Madrid-Barajas, the departure trajectory of an aircraft follows the direction vector $\vec{d} = (12, 5)$ in nautical miles. For traffic separation reasons, an escape maneuver described by a vector $\vec{v} = (x, y)$ must be scheduled satisfying three conditions simultaneously:\n1) It must be strictly perpendicular to the trajectory $\vec{d}$.\n2) Its magnitude must be exactly $39$ nautical miles.\n3) It must have a component in the direction of the prevailing wind $\vec{w} = (1, 2)$, which requires that $\vec{v} \cdot \vec{w} > 0$.\na) Determine the exact Cartesian components of the escape vector $\vec{v}$ after evaluating which of the two possible perpendicular orientations satisfies the constraint imposed by the wind.\nb) Prove analytically that, for any generic non-zero vector $\vec{u} = (u_1, u_2)$, the only two vectors perpendicular to it that have a fixed magnitude $m > 0$ are given by the expression $\vec{p} = \pm m \cdot \dfrac{(-u_2, u_1)}{\sqrt{u_1^2 + u_2^2}}$, justifying each step from the dot product and the concept of a unit vector.

Topic challenge. How should you orient a solar panel to maximize the energy collected in your locality?

1. Data collection: Check the altitude and azimuth of the Sun in your municipality at four different times on a sunny day on the AEMET or PVGIS website. Write down the coordinates of the solar unit vector $\vec{s}(t) = (x(t), y(t), z(t))$ for each measured instant.

2. Modeling with the dot product: Define the unit vector perpendicular to the panel $\vec{n}$, which depends on the tilt angle $\theta$. The relative power collected is given by the dot product $P(\theta) = \vec{s} \cdot \vec{n} = |\vec{s}| |\vec{n}| \cos(\alpha)$ (if it is negative, the Sun is behind the panel and $P = 0$ is taken). Enter the data into GeoGebra or a spreadsheet and calculate $P(\theta)$ for different angles (in steps of $5^\circ$ or $0,1$ rad).

3. Optimization using derivatives: Model the function of total energy collected as a function of the angle $\theta$, $E(\theta)$. Apply derivative exercises: find the instantaneous rate of change $E'(\theta)$ and identify the optimal angle by solving $E'(\theta) = 0$.

Submit a brief technical report with your data, table of values, calculated derivatives, and your reasoned tilt proposal.

Assessment criteria:

• Accuracy in extracting real data and formulating direction vectors.

• Rigorous application of the dot product to model projected energy.

• Skill in calculating and geometrically interpreting derivatives to optimize the angle.

• Clarity and formal justification in the final report.