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 $\vec{v} = (0, 4)$.

Elena has to calculate the angle formed by the vectors $\vec{u} = (1, -1)$ and $\vec{v} = (0, 3)$. On her exam, she writes the following procedure:

"$\vec{u} \cdot \vec{v} = 1 \cdot 0 + (-1) \cdot 3 = -3$. Since the scalar product cannot be negative because it measures geometric lengths, I change the sign to $+3$. Therefore, $\cos\alpha = \frac{3}{\sqrt{2} \cdot 3} = \frac{1}{\sqrt{2}}$, from which $\alpha = 45^\circ$".

1. a) Point out the conceptual error Elena makes regarding the sign of the scalar product and explain its true geometric meaning in relation to the type of angle.
2. b) Correct the solution by calculating the exact value of the angle $\alpha$ formed by both vectors.

Consider the plane vectors $\vec{u} = (5, 2)$ and $\vec{v} = (-1, 4)$.

1. a) Estimate: Without carrying out the numerical multiplication of their components, mentally place both vectors in the corresponding quadrants of the Cartesian plane and reason whether their scalar product $\vec{u} \cdot \vec{v}$ will be positive, negative, or zero, predicting whether the angle they form is acute, obtuse, or right.
2. b) Check: Now calculate the exact numerical value of $\vec{u} \cdot \vec{v}$ and obtain the measure in degrees of the angle $\alpha$ between them (rounding to the nearest tenth). Justify whether the analytical result confirms your initial prediction.

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.

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.

Determine the coordinates of an unknown vector $\vec{w} = (x, y)$ that simultaneously satisfies the following two geometric conditions:

1. It is orthogonal to the vector $\vec{u} = (3, -4)$.

2. Its signed orthogonal projection onto the direction of the vector $\vec{v} = (0, 5)$ is exactly $-3$.

Find the vector $\vec{w}$, calculate its magnitude, and justify with reasoned arguments whether the solution obtained is unique or if there is more than one vector satisfying both conditions.

In a photovoltaic installation in the province of Sevilla, the direction of incidence of solar radiation at a given moment is described by the unit vector $\vec{s} = (0,6;\, 0,8)$, which points towards the Sun. The orientation of a flat panel is described by a unit vector perpendicular to its surface, $\vec{n}$. The fraction of captured power relative to the theoretical maximum is given by the scalar product $\vec{s} \cdot \vec{n}$ (where the value $1$ corresponds to $100\,\%$ efficiency).

Due to mechanical play, the panel ends up oriented with a normal vector $\vec{n} = (0,8;\, 0,6)$.

1. a) Calculate $\vec{s} \cdot \vec{n}$ and determine what percentage of its theoretical maximum power the panel captures in that position.
2. b) Calculate the misalignment angle between the sun's rays and the perpendicular to the panel.
3. c) If an operator manually rotates the panel to an orientation with normal vector $\vec{n}' = (-0,8;\, 0,6)$, will the front face of the panel generate current? Justify your answer by analysing the sign and geometrically interpreting the value of the scalar product $\vec{s} \cdot \vec{n}'$.

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.

The magnitudes of two vectors are known, $|\vec{u}| = 3$ and $|\vec{v}| = 5$, and they are known to form an angle of $60^\circ$ with each other. It is desired to find the cosine of the angle $\theta$ formed by the sum vector $\vec{a} = \vec{u} + \vec{v}$ and difference vector $\vec{b} = \vec{u} - \vec{v}$. Two students propose different approaches:

• Strategy A (Analytic geometry): Set up a Cartesian system with $\vec{u} = (3, 0)$ along the x-axis, determine the components of $\vec{v} = (5\cos 60^\circ, 5\sin 60^\circ)$ using trigonometric ratios, add and subtract to obtain $\vec{a}$ and $\vec{b}$ in coordinates, and apply the analytical formula of the scalar product.

• Strategy B (Vector algebra): Expand symbolically $\vec{a} \cdot \vec{b} = (\vec{u} + \vec{v}) \cdot (\vec{u} - \vec{v})$ and the magnitudes $|\vec{a}|^2 = (\vec{u}+\vec{v})\cdot(\vec{u}+\vec{v})$ and $|\vec{b}|^2 = (\vec{u}-\vec{v})\cdot(\vec{u}-\vec{v})$ using exclusively the commutative and distributive properties and $\vec{x}\cdot\vec{x}=|\vec{x}|^2$, without setting up axes or coordinates.

1. a) Fully carry out both strategies to obtain the exact value of $\cos\theta$.
2. b) Compare the two approaches: which of the two methods reveals a geometric property that does not depend at all on the $60^\circ$ angle? Justify which one is more direct and generalisable to three dimensions.

1. a) Formally prove, using only the algebraic properties of the scalar product (commutative, distributive, and $\vec{x} \cdot \vec{x} = |\vec{x}|^2$), what is known as the Parallelogram law:

1. b) In the port of Algeciras, two tugboats help position a floating platform by pulling via cables hitched to the same towing point. The applied forces are modelled as vectors $\vec{u}$ and $\vec{v}$, with magnitudes $|\vec{u}| = 8\text{ kN}$ and $|\vec{v}| = 6\text{ kN}$. A dynamometer at the hitch records that the total resultant force has a magnitude of $|\vec{u} + \vec{v}| = 12\text{ kN}$. Without calculating the angle formed by the cables or using Cartesian coordinates, apply the identity proven in the previous part to calculate the exact value of the magnitude $|\vec{u} - \vec{v}|$.
2. c) From the data and the previous results, deduce with reasoned justification the sign of the scalar product $\vec{u} \cdot \vec{v}$ and determine whether the angle between the two towing lines is acute, right, or obtuse.