Exercises: Scalar product

A team of engineers is designing the support structure for a telecommunications antenna. They have defined two stay cables represented by the vectors $\vec{u} = (2, -3, 1)$ and $\vec{v} = (4, 2, 5)$. Calculate the scalar product of both vectors to study the angle they form at the central node.

Two surveillance drones are in fixed positions in space. The first drone is at point $A(1, 2, -1)$ and the second at point $B(3, -2, 2)$. If the control centre is at the origin of coordinates $O(0, 0, 0)$, calculate the magnitude of the position vector of each drone, $|\vec{OA}|$ and $|\vec{OB}|$.

Determine the value of the real parameter $k$ such that vectors $\vec{a} = (k, 4, -2)$ and $\vec{b} = (3, -2, 5)$ are perpendicular (orthogonal). This type of calculation is fundamental in video game programming for detecting collisions between surfaces.

At a theme park ride, two cables pull a cabin. The directions of the cables are given by the vectors $\vec{u} = (1, 0, 1)$ and $\vec{v} = (0, 1, 1)$. Using the definition of scalar product, find the angle $\alpha$ between the two cables.

A wind vector has components $\vec{w} = (3, -4, 12)$. In meteorology, it is useful to work with normalised directions. Calculate the unit vector $\vec{u_w}$ in the same direction as $\vec{w}$.

Given the vector $\vec{a} = (6, 2, -3)$, calculate its scalar projection onto the x-axis ($X$-axis). To do this, use the scalar product of $\vec{a}$ with the unit vector of that axis, $\vec{i} = (1, 0, 0)$. Represent the idea of the projection graphically.

You can do it on paper and send a photo with your answer.

Consider the points $A(1, 1, 1)$, $B(3, 1, 1)$ and $C(1, 4, 1)$.

1. a) Find the vectors $\vec{AB}$ and $\vec{AC}$.
2. b) Calculate their scalar product $\vec{AB} \cdot \vec{AC}$.
3. c) Justify whether triangle $ABC$ has a right angle at vertex $A$.

In physics, the work done by a constant force is defined as the scalar product of the force vector and the displacement vector: $W = \vec{F} \cdot \vec{d}$. If a force of $\vec{F} = (5, 3, 2)$ newtons displaces an object from point $P(0, 0, 0)$ to point $Q(2, 4, 1)$ metres, calculate the work done expressed in joules.

Elena has to calculate the scalar projection of vector $\vec{a} = (3, -1, 2)$ onto vector $\vec{b} = (1, 2, -2)$ and deduce whether the angle between them is acute, right, or obtuse. On her exam she writes the following:

“We calculate the projection: $\text{proy}_{\vec{b}}\vec{a} = \frac{\vec{a}\cdot\vec{b}}{|\vec{a}|} = \frac{3\cdot 1 + (-1)\cdot 2 + 2\cdot(-2)}{\sqrt{3^2+(-1)^2+2^2}} = \frac{3-2-4}{\sqrt{14}} = \frac{-3}{\sqrt{14}}$. Since a projection represents a geometric measurement, it cannot be negative, so the correct result is $\text{proy}_{\vec{b}}\vec{a} = \frac{3}{\sqrt{14}}$. Furthermore, being a positive quantity, this shows that the angle formed by $\vec{a}$ and $\vec{b}$ is acute.”

Identify with reasoning the three conceptual errors made by Elena (in the formula used, in the interpretation of the sign of the projection, and in the conclusion about the angle), explain why they are incorrect, and write the properly corrected solution.

Given the vectors $\vec{u} = (4, 1, 1)$ and $\vec{v} = (-1, 5, 2)$:

1. a) Before performing any exact numerical calculation:

• Observe the signs and magnitudes of the corresponding components and estimate with reasoning whether the dot product $\vec{u}\cdot\vec{v}$ will be positive, negative, or very close to zero. From this, predict whether the angle $\alpha$ between them will be acute, obtuse, or nearly a right angle.

• Knowing that $\vec{u}$ has almost all its weight on the $X$ axis and $\vec{v}$ on the $Y$ axis, estimate whether the shadow or scalar projection of $\vec{u}$ onto $\vec{v}$ will be large or small compared to the length of $\vec{v}$.

1. b) Now calculate the dot product $\vec{u}\cdot\vec{v}$, the magnitudes $|\vec{u}|$ and $|\vec{v}|$, the exact angle $\alpha$ (rounded to the nearest degree), and the scalar projection $\text{proy}_{\vec{v}}\vec{u}$.
2. c) Compare the numerical results obtained with your estimates from part a) and reason whether your initial geometric intuition was accurate.

Given the vectors $\vec{u} = (m, 2, -1)$ and $\vec{v} = (3, m, m+2)$, determine the values of the real parameter $m$ such that both vectors are orthogonal. If several values exist, explain whether the resulting vectors in each case have the same magnitude.

Let $\vec{u}$ and $\vec{v}$ be two vectors in space such that their magnitudes are $|\vec{u}| = 4$ and $|\vec{v}| = 3$. Given that the angle between them is $60^\circ$, calculate the value of the scalar product of vectors $\vec{a} = 2\vec{u} - \vec{v}$ and $\vec{b} = \vec{u} + 3\vec{v}$ using the distributive and commutative properties.

Calculate the scalar projection and the vector projection of the vector $\vec{a} = (1, -2, 5)$ onto the vector $\vec{b} = (3, 0, 4)$.

1. a) Calculate the scalar projection of $\vec{a}$ onto $\vec{b}$.

1. b) Calculate the vector projection of $\vec{a}$ onto $\vec{b}$.

1. c) Interpret graphically whether the projection vector has the same direction as $\vec{b}$, based on the sign of the dot product $\vec{a}\cdot\vec{b}$ that you obtained.

The vertices of a triangle in space are $A(1, 2, 1)$, $B(3, -1, 0)$ and $C(-1, 0, 4)$.

• a) Calculate the interior angle at vertex $A$.

• b) Use the scalar product to determine whether the triangle is acute-angled, right-angled or obtuse-angled.

A vector $\vec{w}$ has magnitude 10 and its first two direction cosines are $\cos \alpha = 0,6$ and $\cos \beta = -0,8$.

• a) Determine the third direction cosine, $\cos \gamma$.

• b) Find the components of the vector $\vec{w}$.

In an engineering structure, a force of $\vec{F} = (4, -3, 2)$ newtons acts on a point, displacing it from position $P(1, 1, 1)$ to $Q(5, 2, -3)$ (coordinates in metres). Calculate the work done by the force, recalling that work is the scalar product of the force vector and the displacement vector $\vec{W} = \vec{F} \cdot \vec{d}$.

Determine the value of $k$ so that the angle formed by vectors $\vec{u} = (1, 1, 0)$ and $\vec{v} = (0, 1, k)$ is exactly $45^\circ$. Justify whether the obtained value of $k$ is unique or if there are several possible orientations for vector $\vec{v}$.

Let $\vec{u}$ and $\vec{v}$ be two unit vectors. Prove that the sum vector $\vec{u} + \vec{v}$ and the difference vector $\vec{u} - \vec{v}$ are always perpendicular to each other, regardless of the components of these vectors. Use this result to explain the geometrical relationship between the diagonals of a rhombus.

Determine all vectors $\vec{w} = (x, y, z)$ in space that simultaneously satisfy the following three conditions:

1. It is orthogonal to the vector $\vec{u} = (1, 1, 0)$.

2. It forms an angle of $45^\circ$ with the vector $\vec{v} = (0, 1, 1)$.

3. Its magnitude is $|\vec{w}| = \sqrt{6}$.

Once the vectors are obtained, explain using geometric arguments why the problem admits exactly two distinct solutions (interpret the geometric intersection between the plane of orthogonality, the cone of constant angle, and the sphere of fixed radius).

In a photovoltaic solar energy facility located in La Mancha, the panels are mounted on a fixed structure oriented according to the unit normal vector $\vec{n} = (0; 0,6; 0,8)$ with respect to a reference frame where the positive $X$ semi-axis points East, the positive $Y$ semi-axis points North, and the positive $Z$ semi-axis points towards the zenith (vertically upward). At 14:00 h on a spring day, the solar radiation strikes in the direction of the unit vector $\vec{s} = (0,3; -0,4; -0,866)$ (oriented in the direction of travel of the rays toward the ground).

1. a) Model the direction of the sun rays with respect to the panel: obtain the unit vector pointing toward the Sun, $\vec{u}_{\text{sol}} = -\vec{s}$, and calculate using the dot product the angle of incidence $\theta$ that the rays form with the normal vector $\vec{n}$ of the panel.
2. b) The instantaneous power generated by the installation is modeled as $P = P_{\text{máx}}\cdot \cos\theta$, where $P_{\text{máx}} = 20\text{ kW}$ corresponds to the optimal situation in which the rays strike with complete perpendicularity onto the panel. Calculate the power generated by the installation at that time.
3. c) From the formula of the dot product, justify with reasoning how the panel's normal vector should be reoriented at that time to achieve maximum power.

A team of engineers is calibrating the orientation sensor of an industrial inspection drone. The reference system is fixed at the take-off base. It is known that the main axis of the drone must follow the direction of a vector $\vec{w}$ that simultaneously satisfies the following three conditions:

1. Vector $\vec{w}$ must be a unit vector.

2. $\vec{w}$ must be orthogonal to vector $\vec{u} = (1, -2, 2)$, which represents the direction of the prevailing wind.

3. The angle formed by $\vec{w}$ with the position vector of the target, $\vec{v} = (0, 3, 4)$, must be $45^\circ$.

Determine the coordinates of the possible vectors $\vec{w}$ that the drone's navigation software can use.

Consider a tetrahedron with vertices $O(0,0,0)$, $A(a, 0, 0)$, $B(0, b, 0)$ and $C(0, 0, c)$ with $a, b, c > 0$. Let $M$ be the midpoint of the segment $BC$.

1. a) Using the scalar product exclusively, prove that vector $\vec{AM}$ is perpendicular to segment $BC$ if and only if the legs of the base along axes $Y$ and $Z$ have the same length ($b = c$).

1. b) If we fix $a=4$, $b=2$ and $c=2$, calculate the orthogonal projection of vector $\vec{OA}$ onto the direction defined by vector $\vec{AB}$. Express the result as a vector.

In the design of a state-of-the-art architectural structure, a tension cable is required to connect a fixed point $P(1, 2, 3)$ to a variable point $Q$ located on a metal beam whose path is given by line $r: (x, y, z) = (2, 0, 1) + \lambda(1, -1, 0)$.

It is desired that the cable be as short as possible. Use the concepts of scalar product and projection to:

1. a) Determine the coordinates of point $Q$ that minimises the length of the cable.
2. b) Check that, for this point, vector $\vec{PQ}$ is perpendicular to the direction vector of beam $r$.

Given the vectors $\vec{u} = (m, 1, 1)$ and $\vec{v} = (1, m, 1)$:

1. a) Determine the values of the real parameter $m$ for which the angle formed by $\vec{u}$ and $\vec{v}$ is exactly $60^\circ$.

1. b) For the value of $m=0$, find a vector $\vec{w}$ with magnitude $\sqrt{6}$ that is a linear combination of $\vec{u}$ and $\vec{v}$, and is also orthogonal to vector $\vec{a} = (1, 1, -1)$.

Prove, using the properties of the scalar product, that for any two vectors $\vec{x}, \vec{y} \in \mathbb{R}^3$, the following identity (known as the Polarisation Identity) holds:

Next, use this result to calculate the scalar product of two vectors whose sum and difference have magnitudes $|\vec{x} + \vec{y}| = 10$ and $|\vec{x} - \vec{y}| = 6$. What angle do these vectors form if we also know that their magnitudes are equal?

An artificial satellite moves along a straight trajectory. At a given instant, its position vector is $\vec{r} = (2, 2, 1)$ and its velocity vector is $\vec{v} = (1, -2, 2)$. A control centre on Earth emits a radio signal whose direction is given by vector $\vec{s} = (k, 1, 0)$.

Calculate the value of parameter $k$ so that the projection of the satellite's velocity onto the direction of the radio signal is zero. Interpret physically what it means for this projection to be zero in terms of the satellite approaching or moving away from the signal source.

Let $B = \{ \vec{e}_1, \vec{e}_2, \vec{e}_3 \}$ be a basis of $\mathbb{R}^3$ that is not necessarily orthonormal. The following scalar products between the basis vectors are known:

• $\vec{e}_1 \cdot \vec{e}_1 = 2$, $\vec{e}_2 \cdot \vec{e}_2 = 1$, $\vec{e}_3 \cdot \vec{e}_3 = 3$

• $\vec{e}_1 \cdot \vec{e}_2 = -1$, $\vec{e}_1 \cdot \vec{e}_3 = 0$, $\vec{e}_2 \cdot \vec{e}_3 = 1$

Calculate the magnitude of vector $\vec{u} = 2\vec{e}_1 - 3\vec{e}_2 + \vec{e}_3$ and determine the cosine of the angle that this vector $\vec{u}$ forms with vector $\vec{e}_1$.

Consider the points in space $A(1, 0, -1)$ and $B(3, 2, 1)$. Let $r$ be the line passing through the point $C(0, 4, 3)$ with direction vector $\vec{d} = (1, 1, 1)$.

Determine whether there exists any point $P$ belonging to the line $r$ such that the vectors $\vec{AP}$ and $\vec{BP}$ are orthogonal. Is there a unique solution? Justify your answer by analysing the resulting quadratic equation.

Consider two vectors $\vec{u}$ and $\vec{v}$ in space such that $|\vec{u}| = 3$, $|\vec{v}| = 5$, and the angle between them is $\alpha = 120^\circ$. The linear combination vector $\vec{w} = 2\vec{u} - 3\vec{v}$ is defined. You are asked to find the magnitude of $\vec{w}$ and the angle $\beta$ that $\vec{w}$ forms with $\vec{u}$. To solve it, two strategies are proposed:

• Method 1 (analytical by coordinates): Set a convenient Cartesian frame where $\vec{u} = (3, 0, 0)$ and $\vec{v} = (5\cos 120^\circ, 5\sin 120^\circ, 0) = (-2,5; 2,5\sqrt{3}; 0)$, calculate the components of $\vec{w}$, and apply the usual coordinate formulas.

• Method 2 (intrinsic algebraic): Expand $|\vec{w}|^2 = (2\vec{u}-3\vec{v})\cdot(2\vec{u}-3\vec{v})$ using only the properties of the dot product (commutative, distributive, scalar factoring, and $\vec{a}\cdot\vec{a}=|\vec{a}|^2$), and calculate $\cos\beta = \frac{\vec{w}\cdot\vec{u}}{|\vec{w}|\,|\vec{u}|}$ by expanding the numerator without introducing axes.

1. a) Solve the problem applying both methods until obtaining the exact value of $|\vec{w}|$ and $\cos\beta$.
2. b) Critically compare both methods: which one requires fewer calculation steps and avoids carrying unnecessary roots? What additional difficulty would Method 1 present if you had three non-coplanar vectors in space for which you only know their magnitudes and pairwise angles? Justify which method is conceptually more powerful and generalizable in mathematics.

In a logistics warehouse in Zaragoza, an overhead crane applies a force $\vec{F} = (10, -5, 15)\text{ kN}$ on a structural joint. An anchoring guide pillar follows the direction of the vector $\vec{d} = (1, 2, -2)$.

1. a) Every vector $\vec{F}$ decomposes uniquely as $\vec{F} = \vec{F}_\parallel + \vec{F}_\perp$, where $\vec{F}_\parallel$ is parallel to $\vec{d}$ and $\vec{F}_\perp$ is orthogonal to $\vec{d}$. Deduce with reasoning, by setting $\vec{F}_\parallel = \lambda\vec{d}$ and requiring that $(\vec{F} - \lambda\vec{d})\cdot\vec{d} = 0$, the vector formula $\vec{F}_\parallel = \left(\frac{\vec{F}\cdot\vec{d}}{|\vec{d}|^2}\right)\vec{d}$. Calculate the exact components of $\vec{F}_\parallel$ and $\vec{F}_\perp$.
2. b) Explicitly check that $\vec{F}_\perp\cdot\vec{d} = 0$ and verify that the Pythagorean relation $|\vec{F}|^2 = |\vec{F}_\parallel|^2 + |\vec{F}_\perp|^2$ holds.
3. c) The force $\vec{F}$ is removed and another unknown force $\vec{G}$ with magnitude $|\vec{G}| = 10\text{ kN}$ is applied. Without knowing the components of $\vec{G}$, it is known that it forms an obtuse angle with vector $\vec{d}$ and that the magnitude of its component perpendicular to the cable is $|\vec{G}_\perp| = 6\text{ kN}$. Using the proven vector relationships, calculate the exact numerical value of the scalar projection $\text{proy}_{\vec{d}}\vec{G}$ and the dot product $\vec{G}\cdot\vec{d}$, rigorously justifying the sign obtained.