Vector product - Aetherwisp

The vector product, also called cross product, is an operation between two vectors that produces a third vector that is orthogonal to both. Given two vectors $\mathbf{a}$ and $\mathbf{b}$ , the vector product is defined as

\mathbf{a}\times\mathbf{b}\equiv \lVert \mathbf{a} \rVert \ \lVert \mathbf{b} \rVert \sin\theta\ \hat{\mathbf{n}}

where $\lVert \cdot \rVert$ denotes the norm of a vector, $\theta$ is the angle between the vectors and $\hat{\mathbf{n}}$ is the unit vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$ . The direction of $\hat{\mathbf{n}}$ can be inferred using the right-hand rule. For other forms, see below.

The vector product is defined in a satisfactory manner only in three dimensional spaces, typically $\mathbb{R}^{3}$ . Although there exist generalizations to $N$ -dimensional vector spaces, none of them meet all of the properties below and are consequently not used.

Properties#

The vector product has the following properties:

It is distributive over sums: $\mathbf{a}\times(\mathbf{b}+\mathbf{c})=\mathbf{a}\times\mathbf{b}+\mathbf{a}\times\mathbf{c}$ .
It is anticommutative: $\mathbf{a}\times\mathbf{b}=-\mathbf{b}\times\mathbf{a}$ .
The vector product of a vector with itself is always zero: $\mathbf{a}\times\mathbf{a}=0$ .

Matrix representation#

Given two vectors $\mathbf{a}$ and $\mathbf{b}$ , the vector product $\mathbf{a}\times\mathbf{b}$ can be represented in matrix form. Firstly, define the antisymmetric matrix of $\mathbf{a}$ :

C(\mathbf{a})\equiv\begin{pmatrix}0 & -a_{3} & a_{2} \\ a_{3} & 0 & -a_{1} \\ -a_{2} & a_{1} & 0\end{pmatrix}

Then, the vector product is the matrix multiplication with $\mathbf{b}$ :

\mathbf{a}\times\mathbf{b}\equiv C(\mathbf{a})\mathbf{b}=\begin{pmatrix}0 & -a_{3} & a_{2} \\ a_{3} & 0 & -a_{1} \\ -a_{2} & a_{1} & 0\end{pmatrix}\begin{pmatrix}b_{1} \\ b_{2} \\ b_{3}\end{pmatrix}=\begin{pmatrix}a_{2}b_{3}-a_{3}b_{2} \\ a_{3}b_{1}-a_{1}b_{3} \\ a_{1}b_{2}-a_{2}b_{1}\end{pmatrix}

In Cartesian coordinates#

In Cartesian coordinates in particular, a second matrix representation is possible in the form using a pseudomatrix. Calling $\hat{\mathbf{i}}$ , $\hat{\mathbf{j}}$ and $\hat{\mathbf{k}}$ the unit vectors of the Cartesian triad, the vector product can be calculated as the determinant of the following pseudomatrix:

\mathbf{a}\times\mathbf{b}\equiv\begin{vmatrix}\hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3}\end{vmatrix}

It can be solved, for example, with Sarrus' rule:

\mathbf{a}\times\mathbf{b}=(a_{2}b_{3}-a_{3}b_{2})\hat{\mathbf{i}}+(a_{3}b_{1}-a_{1}b_{3})\hat{\mathbf{j}}+(a_{1}b_{2}-a_{2}b_{1})\hat{\mathbf{k}}

which gives the same result as above.

Tensor representation#

The vector product can also be represented by the Levi-Civita tensor $\epsilon$ . The $i$ -th component of the vector product is

(\mathbf{a}\times \mathbf{b})_{i}=\sum_{j,k=1,2,3} \epsilon_{ijk}a_{j}b_{k}

Interaction with rotations#

The vector product is distributive with respect to a rotation $R$ :

(R\mathbf{a})\times(R\mathbf{b})=R(\mathbf{a}\times\mathbf{b})