The 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 $XY$?

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

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 $R(2, 0, 1)$.

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).

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.

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.