Exercises: Vector product

Given the vectors $\vec{u} = (2, -1, 3)$ and $\vec{v} = (0, 4, 1)$, calculate their vector product $\vec{u} \times \vec{v}$ using the determinant rule with the standard basis vectors $\{\vec{i}, \vec{j}, \vec{k}\}$.

Calculate a vector that is simultaneously perpendicular to $\vec{a} = (1, 1, 1)$ and $\vec{b} = (3, 0, 2)$. Check the result by taking the scalar product of the obtained vector with $\vec{a}$.

Find the area of the parallelogram whose sides are the vectors $\vec{u} = (5, 0, 0)$ and $\vec{v} = (2, 3, 0)$. What relationship do you observe between this result and the area of a rectangle of base 5 and height 3 situated in the plane $XY$?

Determine the area of the triangle whose vertices are the points in space $A(1, 0, 2)$, $B(3, 1, 2)$, and $C(1, 2, 5)$.

Check whether the vectors $\vec{u} = (1, -2, 4)$ and $\vec{v} = (-2, 4, -8)$ are parallel by calculating their vector product. Explain what condition the result of a vector product must satisfy to ensure that two vectors have the same direction.

Given the unit vectors of the coordinate axes $\vec{i} = (1, 0, 0)$, $\vec{j} = (0, 1, 0)$, and $\vec{k} = (0, 0, 1)$, calculate the following vector products:

1. a) $\vec{i} \times \vec{j}$
2. b) $\vec{j} \times \vec{i}$

Considering the results, deduce whether the vector product satisfies the commutative property.

Find a vector that is perpendicular to the plane passing through the points $P(0, 0, 0)$, $Q(1, 2, 3)$, and $R(2, 0, 1)$.

In physics, the moment of a force $\vec{F}$ applied at a point with respect to an origin is given by the cross product $\vec{M} = \vec{r} \times \vec{F}$, where $\vec{r}$ is the position vector of the point of application. Calculate the moment of a force $\vec{F} = (0, 10, 0)$ Newtons applied at the point with coordinates $\vec{r} = (3, 0, 0)$ meters. The figure shows a sketch (not to scale) of $\vec{r}$ and $\vec{F}$.

Lucía has to find the coordinates of a vector perpendicular simultaneously to $\vec{u} = (2, -1, 3)$ and $\vec{v} = (1, 4, -2)$. To solve it, she sets up the determinant with the standard basis vectors and writes on her exam:

After computing, she concludes that the required vector is $\vec{w} = (-10, -7, 9)$.

1. a) Without recalculating the entire cross product, use the dot product to prove that the vector $\vec{w}$ obtained by Lucía is not perpendicular to $\vec{u}$.
2. b) Identify and explain accurately the conceptual or algebraic error that Lucía made in the expansion of the determinant.
3. c) Correct the calculation and write the correct coordinates of the vector $\vec{u} \times \vec{v}$.

In three-dimensional space, consider the vertices of a triangle: $A(0, 0, 0)$, $B(4, 0, 0)$, and $C(2, 5, 1)$ (with coordinates measured in metres).

1. a) Before calculating any determinant or cross product, observe the geometric position of the points: side $AB$ lies on the $X$-axis and vertex $C$ is located at $y = 5$ with a small vertical elevation ($z = 1$). From the area of the projection of the triangle onto the horizontal $XY$-plane, make a reasoned estimate of the area of triangle $ABC$ in space and indicate between which two consecutive integers its value must lie.
2. b) Calculate the exact area of the triangle using the magnitude of the cross product $\frac{1}{2}|\vec{AB} \times \vec{AC}|$.
3. c) Compare the exact value obtained with your previous estimate and explain why the actual area in space is slightly greater than that of the triangle projected onto the horizontal plane.

Given the vectors $\vec{u} = (1, -2, 3)$ and $\vec{v} = (0, 1, 4)$, calculate the coordinates of a vector $\vec{w}$ that is simultaneously perpendicular to $\vec{u}$ and to $\vec{v}$, and has a magnitude equal to $\sqrt{130}$ units. How many possible solutions are there for this vector?

A team of surveyors is measuring a triangular plot on uneven ground. They have determined that the vertices of the triangle, relative to a Cartesian coordinate system in metres, are $A(1, 0, 2)$, $B(3, 4, 0)$, and $C(5, 1, -3)$. Calculate the area of this plot using the vector product.

Determine the value of the real parameter $k$ such that the area of the parallelogram defined by the vectors $\vec{a} = (k, 1, 2)$ and $\vec{b} = (1, k, 0)$ is exactly $\sqrt{21}$ square units.

In the design of an industrial component, it is necessary to know the direction of the line $r$ resulting from the intersection of two planes whose normals are $\vec{n}_1 = (2, -1, 5)$ and $\vec{n}_2 = (1, 3, -2)$. Knowing that the direction vector of the line is perpendicular to both normals, use the vector product to find it.

Using the properties of the vector product, prove that for any pair of vectors $\vec{u}$ and $\vec{v}$ in space, the following equality holds:

Then, verify the result for the vectors $\vec{u} = (1, 0, -1)$ and $\vec{v} = (2, 3, 1)$.

An environmental engineer is calculating the efficiency of a flat solar panel. The panel is defined by the points $P(1, 1, 1)$, $Q(2, 3, 1)$, and $R(1, 2, 4)$. To maximise energy capture, they need to know the unit normal vector to the surface of the panel. Find this vector.

Given the vectors $\vec{a} = (2, -1, 0)$ and $\vec{b} = (1, 2, -2)$, find a third vector $\vec{c}$ such that the set $\{\vec{a}, \vec{b}, \vec{c}\}$ forms an orthogonal basis of space (that is, they are pairwise perpendicular).

In a physics experiment, a charged particle moves in a magnetic field. The moment of the force $\vec{M}$ about the origin is given by the cross product $\vec{M} = \vec{r} \times \vec{F}$, where $\vec{r}$ is the position vector of the point of application and $\vec{F}$ is the applied force. If $\vec{r} = (2, -3, 1)$ meters and the force is $\vec{F} = (4, 1, 2)$ Newtons (the figure is a sketch of the situation):

1. a) Calculate the moment vector $\vec{M} = \vec{r} \times \vec{F}$.

1. b) Calculate its magnitude $|\vec{M}|$ and state its physical unit (N·m).

In the courtyard of a high school in Sevilla, a triangular shade sail is going to be installed to create a shaded area against high temperatures. The canvas is anchored at three support points fixed to the surrounding walls, whose spatial coordinates referred to the horizontal ground ($XY$-plane, in metres) are $A(0, 0, 3)$, $B(4, 0, 4)$, and $C(0, 6, 5)$.

1. a) Model the tensioned surface as the triangle with vertices $A$, $B$, and $C$, and calculate its exact surface area (area in $\text{m}^2$) using the cross product.
2. b) To prevent water from pooling during heavy rain episodes, the slope of the canvas must be controlled. Obtain a vector perpendicular to the surface of the sail and calculate the angle it forms with the vertical $\vec{k} = (0, 0, 1)$, deducing from this the slope of the sail with respect to the horizontal plane of the ground.

The vectors $\vec{u} = (1, 2, 2)$ and $\vec{w} = (-6, 6, -3)$ are given, and the goal is to determine an unknown vector $\vec{v}$ such that $\vec{u} \times \vec{v} = \vec{w}$.

1. a) Explain why it is an obligatory geometric condition that $\vec{u} \cdot \vec{w} = 0$ for a solution to exist, and check numerically whether this condition is satisfied in this case.
2. b) Formally prove that the solution to this vector equation is never unique: if $\vec{v}_0$ is a particular solution, prove that any vector of the form $\vec{v} = \vec{v}_0 + \lambda\vec{u}$ (with $\lambda \in \mathbb{R}$) also satisfies the equality $\vec{u} \times \vec{v} = \vec{w}$.
3. c) If we impose the additional condition that $\vec{v}$ must be strictly perpendicular to $\vec{u}$, set up the corresponding equations and determine the coordinates of the unique vector $\vec{v}$ that satisfies simultaneously $\vec{u} \times \vec{v} = \vec{w}$ and $\vec{u} \cdot \vec{v} = 0$.

Given vectors $\vec{u} = (1, -1, 2)$ and $\vec{v} = (3, 1, -1)$, find a vector $\vec{w}$ that simultaneously satisfies the following three conditions:

1. It is perpendicular to vectors $\vec{u}$ and $\vec{v}$.

2. Its norm or magnitude is exactly equal to $\sqrt{750}$.

3. Its component along the $Z$-axis (z-component) is a negative number.

Consider a triangle whose vertices are points $A(k, 0, 1)$, $B(1, 2, 1)$ and $C(0, -1, k)$, where $k$ is a real parameter.

1. a) Determine the expression for the area of the triangle as a function of $k$.
2. b) Find the values of $k$ for which the area of the triangle is exactly $\frac{3\sqrt{6}}{2}$ square units.

It is desired to design a parallelogram-shaped metallic structural part. The sides are defined by the vectors $\vec{a} = (2, m, 1)$ and $\vec{b} = (1, 2, 2)$.

1. a) Calculate the value of the parameter $m$ knowing that the area of said part is $\sqrt{26}$ square units.
2. b) For the values of $m$ found, are the vectors $\vec{a}$ and $\vec{b}$ perpendicular? Justify your answer without computing the dot product.

In a robotics study, a robot arm exerts a force of $\vec{F} = (2, -3, 1)$ newtons on a bolt situated at the point $P(1, 1, 4)$. The moment of a force about a point $O$ is defined as the vector product $\vec{M} = \vec{r} \times \vec{F}$, where $\vec{r}$ is the position vector of the point of application of the force relative to the centre of rotation.

If the centre of rotation is located at the origin of coordinates $O(0, 0, 0)$, calculate:

1. a) The moment vector $\vec{M}$.
2. b) The area of the triangle whose sides are the vectors $\vec{r}$ and $\vec{F}$.

Given the vectors $\vec{u} = (2, 1, -1)$ and $\vec{v} = (1, 0, 3)$, find all vectors $\vec{x}$ that simultaneously satisfy the following equations:

1. $\vec{x} \times \vec{u} = \vec{u} \times \vec{v}$

2. $\vec{x} \cdot \vec{v} = 0$

(Hint: Use the distributive properties of the vector product and the linearity condition).

Let $V$ be the volume of the parallelepiped defined by the vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$.

If $\vec{a} = (1, 1, 0)$, $\vec{b} = (0, 2, 1)$, and $\vec{c} = (m, 0, 1)$:

1. a) Find the values of $m$ so that the volume is 4 cubic units.
2. b) For each of those values of $m$, calculate the area of the face of the parallelepiped defined by the vectors $\vec{a}$ and $\vec{b}$, and use this information to find the height of the parallelepiped relative to that base.

Analytically prove Lagrange's Identity for the vectors $\vec{u} = (x, 1, 0)$ and $\vec{v} = (0, y, 1)$, verifying that:

Then, use this identity to find the area of the parallelogram formed by $\vec{u}$ and $\vec{v}$ if it is known that $|\vec{u}| = 2$, $|\vec{v}| = 3$, and their dot product is $\vec{u} \cdot \vec{v} = \sqrt{8}$.

A quadrangular solar panel has its vertices at the points $P(1, 0, 0)$, $Q(3, 2, 1)$, $R(2, 5, 3)$, and $S(0, 3, 2)$.

1. a) Check whether the four points are coplanar (lie in the same plane).
2. b) If they are coplanar, calculate the total area of the solar panel by decomposing it into two triangles and using the vector product.

We want to calculate the distance from the point $P(3, 1, -2)$ to the line $r$ that passes through the point $A(1, 0, 1)$ and has direction vector $\vec{d} = (1, 2, -1)$. Two students propose different strategies:

• Alba's method (vector approach): She interprets that the vector $\vec{AP}$ and the direction vector $\vec{d}$ determine a parallelogram in space whose base is $|\vec{d}|$ and whose height is the desired distance, so she applies the formula $d(P, r) = \frac{|\vec{AP} \times \vec{d}|}{|\vec{d}|}$.

• Bruno's method (analytic approach): He finds the general equation of the plane $\pi$ perpendicular to line $r$ that passes through $P$, calculates the intersection point $M$ between line $r$ and plane $\pi$ (orthogonal projection of $P$ onto $r$), and obtains the distance by calculating the magnitude $|\vec{PM}|$.

1. a) Justify in detail, using geometric arguments, why the formula used in Alba's method rigorously provides the minimum distance from $P$ to the line $r$.
2. b) Work through both methods step by step and verify that they lead to exactly the same numerical result.
3. c) Critically compare both approaches: which one is more direct and has a lower risk of algebraic error if you are only asked for the numerical value of the distance? In what subsequent geometric problem would it be essential to have obtained the point $M$ using Bruno's method instead of limiting yourself to Alba's calculation?

Three non-zero vectors $\vec{u}, \vec{v}, \vec{w} \in \mathbb{R}^3$ simultaneously satisfy the following two vector equalities:

1. a) Rigorously prove, using the definition and geometric properties of the cross product, that the three vectors $\vec{u}$, $\vec{v}$, and $\vec{w}$ are pairwise perpendicular.
2. b) Using the magnitude formula $|\vec{a} \times \vec{b}| = |\vec{a}|\cdot|\vec{b}|\cdot\sin\theta$, prove that the magnitude of vector $\vec{v}$ is uniquely determined by these two conditions and calculate its exact value.
3. c) Deduce the mathematical relationship between the magnitudes $|\vec{u}|$ and $|\vec{w}|$.
4. d) Construct a concrete numerical example by proposing simple coordinates for $\vec{u}$ and $\vec{v}$ that satisfy the conditions of the problem statement, calculate the resulting vector $\vec{w}$, and verify that both vector equalities hold exactly.