Cross Product Symbol in LaTeX

Written by Admin in Math symbols

Learn how to write cross product symbol in LaTeX, known also as vector product, with the simplest and clearest method

  1. Cross product command in LaTeX
  2. Unit vector example
  3. Matrix example
  4. Big times symbol

In mathematics, the cross product is used to represent the vector product which is a binary operation between vectors in a 3-dimensional space. In simple words, the cross product, is the product of two vectors that generates a third vector orthogonal to the first two. It is denoted by the (x), a multiplication symbol.

\times is the cross product command in LaTeX

Suppose a and b are vectors, then their cross product is defined in LaTeX by

% Cross product in LaTeX
\documentclass{article}

\usepackage{amsmath}

\begin{document}
$\vec{a} \times \vec{b}$
\end{document}

Output:

Comments:

  • \usepackage{amsmath}: it is placed in our preamble (area before \begin{document}). It takes the argument amsmath which is capable to interpreting and facilitating the writing of mathematical symbols, formulas and improves the quality of the mathematics output in our document.
  • \times command which returns x, the cross product symbol.
  • \vec{} command takes a single character as argument and return the character with an arrow mark on top of it.

Definition of cross(vector) product

1. Cross product in the form of a unit vector

The cross product of two vectors, a and b, is defined as follows:

and in LaTeX:

% Vector product LaTeX
\documentclass{article}

\usepackage{amsmath}

\begin{document}
$\vec{a} \times \vec{b} = |a||b|\sin\theta\hat{n}$
\end{document}

Where θ is the angle between the two vectors, and n is the unit vector perpendicular a and b.

The LaTeX commands for sin is \sin, and for θ we use \theta. The \hat{ } command takes a single character as argument and return it with a caret(circumflex) on top of it.

2. Cross Product in the form of a Matrix

In order for you to define the cross product in form of a matrix, you need to know how to typeset matrices in LaTeXOpens in a new tab.. There are several different types of matrices but in this tutorial, we will focus on 2 types:

  • Parenthesized matrix (matrix with parenthesis).
  • Vertical bar matrix (often used to represent determinants).

The parenthesized matrix is obtained in LaTeX with the help of the pmatrix environment and the vertical bar matrix with the help of the vmatrix environment. Take a look at the two syntaxes below:

$\begin{pmatrix}
. . . \\
. . . 
\end{pmatrix}$
$\begin{vmatrix}
. . . \\
. . . 
\end{vmatrix}$

If there are multiple entries, then the row-entries are separated (spaced) using (&) and column entries with (\\) line break command as you will see in our various definitions below.

Example 1:

and here is the corresponding LaTeX code:

Given $\vec{a} = \begin{pmatrix} a_1\\a_2\\a_3 \end{pmatrix}$ and
$\vec{b} = \begin{pmatrix} b_1\\b_2\\b_3 \end{pmatrix}$.\\
Then their cross product $\vec{a} \times \vec{b}$ is defined
$\vec{a} \times \vec{b} = (a_2b_3-a_3b_2)i + (a_3b_1-a_1b_3)j+(a_1b_2-a_2b_1)k$

Example 2:

This is obtained by the following LaTeX code:

% determinant form
\begin{align*}
\vec{a} \times \vec{b} =\begin{vmatrix}
i & j & k\\
a_1 & a_2 & a_3\\
b_1 & b_2 & b_3
\end{vmatrix} = (a_2b_3-a_3b_2)i + (a_3b_1-a_1b_3)j+(a_1b_2-a_2b_1)k
\end{align*}

Big times symbol in LaTeX

Big cross product symbol can be obtained in LaTeX using the command \bigtimes provided by the mathabx package. Here is an example: 

% Big times symbol
\documentclass{article}
\usepackage{mathabx}

\begin{document}
$\vec{a} \bigtimes \vec{b}$
\end{document}

Conclusion

This tutorial gives a brief description of the concept of cross(vector) product and at the same time gives an in depth knowledge of LaTeX; basically, knowledge on how to generate the mathematics involved in dealing with cross products. Most often in LaTeX, there are multiple ways of achieving same results and this tutorial teaches you the simplest and clearest method to typeset cross products.

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