Cross 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 $\{\vec{i}, \vec{j}, \vec{k}\}$.

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 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 $XY$?

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.

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.

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

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?

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 $\vec{v} = (2, 3, 1)$.

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).

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 $\vec{u} \cdot \vec{v} = \sqrt{8}$.

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

In physics, the moment of a force $\vec{F}$ applied at a point about an origin is given by the vector product $\vec{M} = \vec{r} \times \vec{F}$, where $\vec{r}$ is the position vector. Calculate the moment of a force of $\vec{F} = (0, 10, 0)$ newtons applied at the point with coordinates $\vec{r} = (3, 0, 0)$ meters.

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 meters, are $A(1, 0, 2)$, $B(3, 4, 0)$, and $C(5, 1, -3)$. Calculate the area of this plot using the vector product.

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 maximize energy capture, they need to know the unit normal vector to the surface of the panel. Find this vector.

In a physics experiment, a charged particle moves in a magnetic field. The moment of the magnetic force $\vec{M}$ about the origin is given by the vector product $\vec{M} = \vec{r} \times \vec{F}$, where $\vec{r}$ is the position vector and $\vec{F}$ is the applied force. If $\vec{r} = (2, -3, 1)$ meters and the force is $\vec{F} = (4, 1, 2)$ newtons, calculate the moment vector $\vec{M}$ and its physical magnitude (magnitude).

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 center of rotation.

If the center 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 $\vec{F}$.

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.

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.

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.

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).