Exercises: Scalar triple product

Given the vectors $\vec{u} = (2, -1, 3)$, $\vec{v} = (0, 4, 1)$, and $\vec{w} = (1, 1, -2)$, calculate the value of the scalar triple product $[\vec{u}, \vec{v}, \vec{w}]$ using the definition based on the determinant of their components.

An engineering student is designing a component using CAD software. The base of the component is defined by the vectors $\vec{a} = (3, 0, 0)$ and $\vec{b} = (0, 5, 0)$, while the third vector defining the depth is $\vec{c} = (1, 2, 4)$. Calculate the volume of the parallelepiped formed by these three vectors.

Check, giving reasons, whether the vectors $\vec{a} = (1, 2, -1)$, $\vec{b} = (3, 0, 2)$, and $\vec{c} = (4, 2, 1)$ are coplanar. To do this, calculate their scalar triple product and justify your answer based on the result obtained.

Given the vectors $\vec{u} = (1, 0, 1)$ and $\vec{v} = (2, 1, -1)$, find a third vector $\vec{w}$ of the form $(0, 0, z)$ such that the volume of the parallelepiped defined by the three is exactly 5 cubic units.

Given the lines $r$ and $s$ defined by the following equations:

$r: \begin{cases} x + y - z = 1 \\ 2x - z = 3 \end{cases}$

$s: \frac{x-1}{2} = \frac{y+1}{1} = \frac{z}{-1}$

Determine the equation of the line $t$ that is the common perpendicular to $r$ and $s$. To do this, first find the direction vectors of both lines and justify, giving reasons, that they are skew lines in space.

Study, depending on the real parameter $a$, the relative position of the lines $r$ and $s$:

$r: \frac{x-1}{a} = \frac{y-2}{-1} = \frac{z}{2}$

$s: \begin{cases} x + 2y = 3 \\ ax + z = 1 \end{cases}$

In the particular case where the lines intersect, calculate the coordinates of the intersection point $P$ and the equation of the plane containing them.

Consider the point $P(1, 0, -1)$ and the lines:

$r: x = y = z$

$s: \begin{cases} x = 3 + t \\ y = 3 + 2t \\ z = 1 + t \end{cases}$

Find the general equation of the plane $\pi$ that passes through point $P$ and is parallel to both lines. Then, determine the equation of a third line $k$ that passes through point $P$, is contained in said plane $\pi$, and is perpendicular to line $r$.

Find the equation of the line $t$ passing through the point $A(1, 2, 1)$ and intersecting the lines $r$ and $s$:

$r: \frac{x}{2} = \frac{y-1}{1} = \frac{z}{1}$

$s: \begin{cases} x + z = 3 \\ y = 2 \end{cases}$

Explain the method used based on the intersection of planes (pencil of planes).

It is known that the line $r$ passes through the point $A(1, m, 0)$ and has direction vector $\vec{u} = (1, 1, a)$. On the other hand, the line $s$ is the intersection of the planes:

$\pi_1: x - y + z = 0$

$\pi_2: 2x + y = 1$

Determine whether there exist values of the parameters $m$ and $a$ such that lines $r$ and $s$ are the same (coincident lines) and, if so, find them.

Two aircraft follow straight-line trajectories given by:

$r: \begin{cases} x = 1 + \lambda \\ y = 2 - \lambda \\ z = 3 + 2\lambda \end{cases}$

$s: \frac{x-a}{1} = \frac{y}{1} = \frac{z-1}{2}$

1. Calculate the value of $a$ for the aircraft trajectories to intersect at a point.

2. For that value of $a$, find the equation of the line $n$ that is perpendicular to the plane formed by the trajectories $r$ and $s$ and passes through the point of collision.

In a Geometry exam, Laura must calculate the volume of the tetrahedron with vertices $O(0, 0, 0)$, $A(2, 0, 0)$, $B(0, 3, 0)$, and $C(1, 2, -4)$. In her solution, she writes the following:

“I calculate the scalar triple product of the vectors $\vec{OA}$, $\vec{OB}$, and $\vec{OC}$ using the determinant:

Since the determinant gives $-24$, the tetrahedron cannot exist with these vertices because a physical volume can never be negative. Therefore, the problem statement contains a geometric inconsistency.”

1. a) Identify and explain with reasoned arguments the two conceptual errors Laura makes: one regarding the interpretation of the sign of the scalar triple product and its relationship with the orientation of the base, and another regarding the direct metric relationship between the scalar triple product and the volume of the tetrahedron.
2. b) Correct the procedure and calculate the real and correct volume of this tetrahedron.

Consider the vectors $\vec{u} = (4, 0, 0)$, $\vec{v} = (0, 5, 0)$, and $\vec{w} = (18, -25, 3)$, which define the three concurrent edges of a parallelepiped.

1. a) Estimate: Without setting up determinants or performing prior algebraic calculations, observe the base of the parallelepiped resting on the $xy$-plane (formed by $\vec{u}$ and $\vec{v}$) and the vertical coordinate given by the third component of $\vec{w}$. Deduce what the volume of the solid must be, reasoning from Cavalieri's principle and the basic formula $V = \text{base area} \cdot \text{height}$. Do the components $18$ and $-25$ affect the height of the parallelepiped above this base?
2. b) Verify: Formally calculate the volume using the absolute value of the scalar triple product $|[\vec{u}, \vec{v}, \vec{w}]|$. Compare the result with your estimate and justify why the slant of edge $\vec{w}$ does not alter the interior capacity of the solid.

It is desired to install a fibre optic cable connecting point $A(1, 4, -2)$ to point $B(3, -1, 0)$. Find the equation of the line passing through both points in symmetric form.

Given the line $r$ expressed in parametric equations:

Determine, giving reasons, whether the points $A(3, -3, 3)$ and $B(1, 0, 5)$ lie on this line.

Express in parametric form the line $s$ defined as the intersection of the following planes (implicit equations):

Find a point and a direction vector of the line $r$ whose symmetric equation is:

An architectural firm designs a skylight in the shape of a tetrahedron whose vertices are the points $A(1, 1, 1)$, $B(3, 2, 0)$, $C(2, 4, -1)$, and $D(0, 0, z)$. Calculate the value of the z-coordinate $z$ knowing that the volume of the skylight is $5$ cubic units and that vertex $D$ lies on the negative semi-axis of the $Z$.

Check whether the points $P(1, 2, -1)$, $Q(3, 0, 2)$, $R(2, 1, 4)$, and $S(5, -2, 7)$ are coplanar using the scalar triple product. If they are not, calculate the volume of the tetrahedron that has these four points as vertices.

Given three linearly independent vectors $\vec{u}$, $\vec{v}$, and $\vec{w}$ such that their scalar triple product is $[\vec{u}, \vec{v}, \vec{w}] = 4$, calculate, giving reasons and applying the properties of determinants, the value of the scalar triple product of the vectors $\vec{x} = 2\vec{u}$, $\vec{y} = \vec{u} + \vec{v}$, and $\vec{z} = \vec{w} - \vec{v}$.

Consider a storage hangar in the shape of a parallelepiped where three of its edges meeting at a vertex are the vectors $\vec{u} = (3, 0, 0)$, $\vec{v} = (0, 4, 0)$, and $\vec{w} = (1, 2, 5)$.

1. a) Calculate the total volume of the hangar.
2. b) If the hangar is divided by an inclined plane passing through the endpoints of the three previous vectors to create a triangular mezzanine (tetrahedron with the origin), what fraction of the total volume of the hangar would this mezzanine occupy?

A tetrahedron-shaped container is to be designed whose three base vertices are fixed: $A(1, 1, 0)$, $B(3, 1, 0)$, and $C(1, 4, 0)$. The upper vertex $D(x, y, z)$ must be chosen in space satisfying two technical conditions:

1. The volume of the tetrahedron must be exactly $4\text{ u}^3$.

2. Vertex $D$ must lie on the directrix line $r$ given by the parametric equations:

1. a) Express the scalar triple product $[\vec{AB}, \vec{AC}, \vec{AD}]$ as a function of the parameter $t$ and describe geometrically what locus all points $D$ in space would form that provide such a volume if they were not restricted to lying on line $r$.
2. b) Find the exact coordinates of all possible points $D$ on line $r$ that satisfy the requested volume.

At a ski resort in Sierra Nevada (Granada), technicians from the weather service install a tensioned canvas canopy to protect instruments from blizzards. The structure has the shape of an irregular tetrahedron whose four vertices are located at the ground anchor $O(0, 0, 0)$ and at the tips of three poles at $A(2, 1, 0)$, $B(0, 3, 1)$, and $C(1, 1, 3)$ (measurements in meters).

1. a) Model the structure using vectors and calculate the usable air capacity (interior volume) protected under the canopy using the scalar triple product.
2. b) To prevent accumulated snow from blocking the sensors, the technicians decide to replace pole $C$ with a vertical one $C'(1, 1, z_0)$ aligned along the same vertical line, aiming to increase the protected volume by exactly $50\,\%$ compared to the initial design. Determine the required elevation $z_0$ and interpret physically whether there is any valid solution that places anchor $C'$ at a lower height than the original ($z_0 < 3$).

The trajectory of a laser beam in an entertainment venue is given by the intersection of the planes $\pi_1: x + y - z = 3$ and $\pi_2: 2x - y + z = 0$.

1. a) Express the equation of the trajectory of the laser in symmetric form.
2. b) A safety sensor is located at the point $S(1, 2, 0)$. Determine whether the laser beam will pass exactly through the position of the sensor.

In the design of an architectural structure, a metal beam lies along the line $r$ passing through the point $A(1, 0, -1)$ and parallel to the line $s: \begin{cases} 3x - y + z = 0 \\ x + y = 2 \end{cases}$.

Obtain the equation of the line $r$ in implicit form (as the intersection of two planes) and calculate a point on it whose $z$-coordinate is equal to $5$.

Given the line $r: \frac{x - 3}{2} = \frac{y + 1}{1} = \frac{z}{2}$:

1. a) Find the equation of a line $s$ passing through the origin of coordinates and parallel to $r$.
2. b) Determine the distance separating the point $Q(1, 1, 1)$ from the line $s$ using a general point on it.

Consider a triangle in space whose vertices are $A(1, 1, 0)$, $B(0, 2, 1)$ and $C(2, 0, 1)$.

1. a) Find the equation of the line containing the median from vertex $A$.
2. b) Find a point $D$ on this line such that the vector $\vec{AD}$ has magnitude $\sqrt{6}$.

Consider the tetrahedron with vertices $A(1, 0, 1)$, $B(k, 1, 1)$, $C(1, k, 0)$, and $D(1, 1, k)$.

1. a) Determine the values of the real parameter $k$ for which the volume of the tetrahedron is $2$ cubic units.
2. b) For $k = 2$, find the height of the tetrahedron from vertex $D$ onto the face formed by vertices $A$, $B$, and $C$.

A team of engineers is designing a support component for a satellite in the shape of a parallelepiped. The base is defined by the vectors $\vec{u} = (2, -1, 0)$ and $\vec{v} = (1, 2, 1)$. The third vector defining the component is $\vec{w} = (k, 1, k+1)$.

1. a) Given that the projection of vector $\vec{w}$ onto the vector product $\vec{u} \times \vec{v}$ has a length of $3$ units, determine the value of $k$.
2. b) Deduce whether there is any value of $k$ that makes the component flat (zero volume).

Using the properties of the scalar triple product, prove vectorially that for any three vectors $\vec{a}, \vec{b}, \vec{c} \in \mathbb{R}^3$, the following holds:

Interpret the result geometrically by comparing the volume of the original parallelepiped with that formed by the sums of its direction vectors.

Consider four points in space: $P(1, 1, 1)$, $Q(2, 3, a)$, $R(k, 1, 2)$, and $S(0, 0, 3)$.

1. a) Determine the relation that $a$ and $k$ must satisfy for the four points to be coplanar.
2. b) If we fix $k = 2$, what value must $a$ take so that the volume of the tetrahedron $PQRS$ is minimal? Justify your answer based on the linear dependence of the vectors.

In a modern architectural structure, a column lies along the common perpendicular to lines $r$ and $s$. Line $r$ passes through the origin and has direction $\vec{d_r} = (1, 0, 1)$. Line $s$ passes through $P(0, 1, 0)$ and has direction $\vec{d_s} = (0, 1, 1)$.

Calculate the volume of the parallelepiped defined by the direction vectors $\vec{d_r}$, $\vec{d_s}$, and a third vector $\vec{v}$ joining any point on $r$ to any point on $s$. Does this volume depend on the points chosen on the lines?

Let $\vec{u}$ be a unit vector and $\vec{v}, \vec{w}$ two vectors such that the area of the parallelogram they form is equal to $10$. Given that the scalar triple product is $[\vec{u}, \vec{v}, \vec{w}] = 5$, calculate the angle that vector $\vec{u}$ makes with the plane containing vectors $\vec{v}$ and $\vec{w}$.

Let $\vec{u}$, $\vec{v}$, and $\vec{w}$ be three linearly independent vectors in space such that their scalar triple product is $[\vec{u}, \vec{v}, \vec{w}] = 5$. Three new vectors are constructed via linear combinations: $\vec{a} = 2\vec{u} - \vec{v}$, $\vec{b} = \vec{u} + \vec{w}$, and $\vec{c} = 3\vec{v} - 2\vec{w}$. To calculate the scalar triple product $[\vec{a}, \vec{b}, \vec{c}]$, two students propose different approaches:

• Strategy 1 (Fictitious basis): Assign convenient numerical components to the original vectors as if they formed a modified canonical orthogonal trihedron, for example $\vec{u} = (1, 0, 0)$, $\vec{v} = (0, 1, 0)$, and $\vec{w} = (0, 0, 5)$, calculate the resulting coordinates of $\vec{a}, \vec{b}, \vec{c}$, and evaluate their $3 \times 3$ determinant.

• Strategy 2 (Determinant algebra): Express the transition matrix $M$ such that $(\vec{a}, \vec{b}, \vec{c}) = (\vec{u}, \vec{v}, \vec{w}) \cdot M$ and use the operational properties of the determinant of a matrix product: $[\vec{a}, \vec{b}, \vec{c}] = \det(M) \cdot [\vec{u}, \vec{v}, \vec{w}]$.

1. a) Rigorously analyze the mathematical validity of Strategy 1: why does it work numerically for calculating the scalar triple product despite unjustifiably assuming that the vectors are perpendicular? What would happen if that same fictitious basis were used to calculate the magnitude of $\vec{a}$ or the angle between $\vec{a}$ and $\vec{b}$?
2. b) Apply Strategy 2 to determine the exact value of $[\vec{a}, \vec{b}, \vec{c}]$, deducing in detail the determinant of the coefficient matrix.
3. c) Justify with reasoned arguments which of the two strategies demonstrates formal mathematical proficiency at the 2nd year of Bachillerato level.

In a slope stabilization project for a railway track alignment in Asturias, a system of three steel tieback anchors is planned, starting from the same foundation point $O(0, 0, 0)$ on the slope and fixed at points $A(2, 0, 1)$, $B(1, 3, 2)$, and a third point $C$ (coordinates in meters).

The geotechnical specifications require three conditions to be met:

1. Point $C$ must rest on the capping beam of the wall, whose directrix is the line $r: \frac{x - 4}{1} = \frac{y - 2}{2} = \frac{z - 1}{1}$.

2. The three tiebacks cannot be coplanar, since a zero volume would imply the loss of three-dimensional stability against lateral thrusts.

3. The virtual tetrahedron bounded by the tiebacks $OABC$ must have a volume of exactly $3\text{ m}^3$ to leave sufficient space for the drainage pipes.

1. a) Model the coplanarity condition using the scalar triple product as a function of the parameter of line $r$ and determine if there is any critical point on that line where the structure would collapse due to coplanarity.
2. b) Find the coordinates of all possible attachment points $C$ located on line $r$ that satisfy the required drainage volume of $3\text{ m}^3$.
3. c) If technical regulations mandate that the anchor height of that tieback must be at an elevation $z \ge 3\text{ m}$ above the base plane to prevent flooding, explain which of the solutions found is the only feasible one.

Unit challenge. Solar optimization of your roof.

Imagine you are going to install a solar panel on your building. Design a technical proposal following these phases:

1. Data collection: Go to Google Earth or the Cadastre's electronic office and obtain the coordinates $(x, y, z)$ in meters of three vertices of your roof ($A$, $B$, and $C$). Define the direction vectors $\vec{u} = \vec{AB}$ and $\vec{v} = \vec{AC}$.

2. Normal vector and basis: In GeoGebra 3D, calculate the normal vector $\vec{n} = \vec{u} \times \vec{v}$ to determine the orientation and the surface area. Check the AEMET website for the solar elevation angle in your locality to define a unit solar ray vector $\vec{s}$, and justify analytically whether $\{\vec{u}, \vec{v}, \vec{s}\}$ form a basis of $\mathbb{R}^3$.

3. Metric analysis: Calculate the solar incidence angle using the dot product between $\vec{n}$ and $\vec{s}$. Use the scalar triple product $[\vec{u}, \vec{v}, \vec{s}]$ to calculate the volume of the intercepted radiation prism.

4. Final report: Prepare a brief report with screenshots of the model in GeoGebra, the algebraic calculations developed step by step, and your assessment of whether the current orientation makes good use of sunlight.

The following will be assessed:

• Rigor in obtaining real data and its three-dimensional vector modeling.

• Correct and justified application of linear independence, bases, dot product, cross product, and scalar triple product.

• Coherent geometric interpretation of the results in the proposed context.