The cross product is one of the most basic and essential operations in the three-dimensional real vector space \(\mathbb{R}^3\). The cross product has significant applications in many fields of science such as physics and computer programming. On the other hand, it is used in many fields of mathematics like differential geometry.

Origins of the cross product formula are rarely explained. In this post, we will try to justify the main formula of the cross product and explain its computation and mathematical applications.

### The cross product formula

Let \(u = (u_1, u_2, u_3)\) and \(v = (v_1, v_2, v_3)\) be vectors in the three-dimensional real vector space \(\mathbb{R}^3\). One interesting question is how to find a vector \(w = (x,y,z)\) perpendicular to \(u\) and \(v\) if we know that \(u\) and \(v\) are not parallel. Without loss of generality, we may assume that \(z \neq 0\), i.e., the vector \(w\) does not lie on the plane \(XY\). Since \(w\) is perpendicular to \(u\) and \(v\) if and only if \(rw\) is so, for each nonzero real number \(r\), we may assume, from the beginning, that \(w = (x_0,y_0,1)\).

The assumption that \(w\) is orthogonal to \(u\) and \(v\) gives the following system of linear equations: $$\begin{cases} u_1 x_0 + u_2 y_0 = – u_3 \\ v_1 x_0 + v_2 y_0 = – v_3 \end{cases}$$ and by solving the system, we find out that $$\begin{cases} x_0 = \displaystyle \frac{u_2 v_3 – u_3 v_2}{u_1 v_2 – u_2 v_1} \\ \\ y_0 = \displaystyle \frac{u_1 v_3 – u_3 v_1}{u_2 v_1 – u_1 v_2} \end{cases}$$

By multiplying the obtained vector by \(u_1 v_2 – u_2 v_1\), we see that we can consider the vector \(w\) with the following components: $$\begin{cases} x = u_2 v_3 – u_3 v_2 \\ y = – (u_1 v_3 – u_3 v_1) \\ z = u_1 v_2 – u_2 v_1\end{cases}$$

In other words, we can symbolically consider the vector $$w = xi + yj + zk$$ to be the determinant of the follwong \(3 \times 3\) matrix, i.e. $$ w = \begin{vmatrix} i & j & k \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}$$

The vector \(w\) obtained in above is called the cross product of the vector \(u\) by \(v\) and denoted by \(u \times v\).

#### Computation and properties of the cross product

One can easily compute the cross product of two vectors using the determinant appearing in the cross product formula.

**Example**. Compute the cross product of \(u\) by \(v\), where $$u = (1,2, 4)$$ and $$v = (2,1,-1).$$

**Solution**. The reader should be able to reach the following answer: $$ u \times v = (-6, 9, -3).$$

**Properties of the cross product**. Let \(u\), \(v\), and \(w\) be three-dimensional real vectors. Some of the basic properties of the cross product are:

- The cross product is not associative. For example, $$(i \times j) \times j \neq i \times (j \times j).$$
- The cross product is not commutative, but in fact, anticommutative, i.e., $$ u \times v = -(v \times u).$$
- \(ru \times su = \vec{0}\), for all real numbers \(r\) and \(s\).
- The cross product distributes over the addition of vectors from both sides, i.e., $$ u \times (v+w) = u \times v + u \times w,$$ and $$(v+w) \times u = v \times u + w \times u.$$
- $$ (ru) \times v = u \times (rv) = r (u \times v).$$

#### Applications of the cross product

One application of the cross product is the following (as we explained above):

The cross product \(u \times v\) of the vectors \(u\) and \(v\) is perpendicular to \(u\) and \(v\).

**Exercise**. Find the equation of a place passing through the following points: $$(1,-2,1),$$ $$(4,-2,-2),$$ and $$(4,1,4).$$

The other application is as follows:

The mixed product (also called the scalar triple product) of the vectors \(u\), \(v\), and \(w\) is defined as the scalar $$u \cdot (v \times w).$$ Note that the absolute value of the mixed product of the vectors \(u\), \(v\), and \(w\) is the volume of the parallelepiped defined by the mentioned three vectors.

**Exercise**. Find the volume of the parallelepiped defined by the following three vectors: $$u = (1,5,-5),$$ $$v = (2,4,3),$$ and $$w = (0,5,-4).$$

**Exercise**. Fix a three-dimensional vector \(v = (v_1, v_2, v_3)\) and define a function \(f\) from \(\mathbb{R}^3\) into \(\mathbb{R}^3\) as follows: $$f(u) = v \times u.$$

- Prove that \(f\) is a linear function.
- Find the matrix of the linear function \(f\).
- Evaluate the kernel of \(f\).

**Exercise**. By direct computation, prove that if \(u\) and \(v\) are three-dimensional real vectors, we have $$\Vert u \times v \Vert^2 + \Vert u \cdot v \Vert^2 = \Vert u \Vert^2 + \Vert v \Vert^2.$$ Using the formula $$ u \cdot v = \Vert u \Vert \Vert v \Vert \cos \theta,$$ where \(\theta\) is the angel between the vectors \(u\) and \(v\), show that $$\Vert u \times v \Vert = \Vert u \Vert \Vert v \Vert \sin\theta.$$

Also, check the post on matrix operations.