# Representation of Cross Product: A tikz Approach

How can I produce the following graph in tikz:

• It is basically produced with segment lines that can be drawn with \draw (x,y) -- (z,w); where x,y,z,w are the coordinates with the points. Also options should be used to change the line style, for example \draw[dashed] – Sigur Aug 17 '13 at 23:33
• It is just the floating ranges that I don't see how to make. – Loie Benedicte Aug 17 '13 at 23:35
• General solution would be nice if it was explained. – Loie Benedicte Aug 17 '13 at 23:36

## 3 Answers

That’s quite easy, I’ll show a way using different “varaibles”. This make the code a little more complex but also more flexible.

Step 1
Set up the basic environment: load the package and the arrows and calc library:

\documentclass{article}% ... or whatever you like

\usepackage{tikz}
\usetikzlibrary{arrows,calc}

\begin{document}
\begin{tikzpicture}
% picture commands
\end{tikzpicture}
\end{document}


Every thing that follows goes inside the {tikzpicture}.

Step 2
Set up the variables. We’ll set up the vectors as coordinates, starting from the origin.

\coordinate (A) at (3,0);
\coordinate (B) at (0,-2);


and the upper left edge of the cross product

\coordinate (cross prod) at (1,4);


We add a simply \def to store the length of the ticks

\def\tick{0.1}


Step 3
Draw the axes: Use the \draw command with the option -> to add the arrow tips. We us a node with some options at the end of the path to add the labels.

\draw [->] (-1,0) -- (6,0) node [below] {$x$};
\draw [->] (0,-1) -- (0,5) node [left]  {$y$};


Then we use a \foreach loop to add the ticks.

\foreach \x in {1,...,5}
\draw (\x,\tick/2) -- ++(0,-\tick) node [below] {\x};
\foreach \y in {1,...,4}
\draw (\tick/2,\y) -- ++(-\tick,0) node [left] {\y};


Step 4
Draw the vectors. We define a style to make all vectors look the same and add it as option to {tikzpicture}. The vectors should be thick, and have “arrow” tips on both sides ([-))

\begin{tikzpicture}[
vec/.style={thick,[-)},
]


Now we can draw the vectors. Starting at a certian point and then using the ++ syntax to add the components of the vector. Again we use a node to add the names.

\draw [vec] (1,-1) -- ++(A) node [midway,below] {$A$};
\draw [vec] (-1,4) -- ++(B) node [midway,left]  {$B$};


Step 5

Now draw the cross product. First we draw the filling in gray using the \fill macro and the rectangle path operation.

\fill [gray] (cross prod) rectangle ++($(A)+(B)$);


An then the borders and the label using \draw again. The -| path operation allows the automatic creation of the right angle.

\draw [thick] ($(cross prod)+(A)$) -| ($(cross prod)+(B)$);
\draw [thick,dashed] ($(cross prod)+(A)$) |- ($(cross prod)+(B)$)
node [pos=0.25,right] {$A \times B$};


And that’s it.

full code

\documentclass{article}% ... or whatever you like

\usepackage{tikz}
\usetikzlibrary{arrows,calc}

\begin{document}
\begin{tikzpicture}[
vec/.style={thick,[-)},
]
% Step 2
\coordinate (A) at (3,0);
\coordinate (B) at (0,-2);
\coordinate (cross prod) at (1,4);
\def\tick{0.2}
% Step 3
\draw [->] (-1,0) -- (6,0) node [below] {$x$};
\draw [->] (0,-1) -- (0,5) node [left]  {$y$};
\foreach \x in {1,...,5}
\draw (\x,\tick/2) -- ++(0,-\tick) node [below] {\x};
\foreach \y in {1,...,4}
\draw (\tick/2,\y) -- ++(-\tick,0) node [left] {\y};
% Step 4
\draw [vec] (1,-1) -- ++(A) node [midway,below] {$A$};
\draw [vec] (-1,4) -- ++(B) node [midway,left]  {$B$};
% Step 5
\fill [gray] (cross prod) rectangle ++($(A)+(B)$);
\draw [thick] ($(cross prod)+(A)$) -| ($(cross prod)+(B)$);
\draw [thick,dashed] ($(cross prod)+(A)$) |- ($(cross prod)+(B)$)
node [pos=0.25,right] {$A \times B$};
\end{tikzpicture}
\end{document}


## update

with some adjustments this will work with every two vectors.

\documentclass{article}% ... or whatever you like

\usepackage{tikz}
\usetikzlibrary{arrows,calc}

\begin{document}
\begin{tikzpicture}[
vec/.style={thick,[-)},
]
% Step 2
\coordinate (A) at (3,-1);
\coordinate (B) at (-0.5,-2);
\coordinate (cross prod) at (1,4);
\def\tick{0.2}
% Step 3
\draw [->] (-1,0) -- (6,0) node [below] {$x$};
\draw [->] (0,-1) -- (0,5) node [left]  {$y$};
\foreach \x in {1,...,5}
\draw (\x,\tick/2) -- ++(0,-\tick) node [below] {\x};
\foreach \y in {1,...,4}
\draw (\tick/2,\y) -- ++(-\tick,0) node [left] {\y};
% Step 4
\draw [vec] (1,-1) -- ++(A) node [midway,below] {$A$};
\draw [vec] (-1,4) -- ++(B) node [midway,left]  {$B$};
% Step 5
\fill [gray] (cross prod) -- +(A) -- +($(A)+(B)$) -- +(B);
\draw [thick] ($(cross prod)+(A)$) -- (cross prod) -- ($(cross prod)+(B)$);
\draw [thick,dashed] ($(cross prod)+(A)$) -- ($(cross prod)+(A)+(B)$)
node [midway,right] {$A \times B$} -- ($(cross prod)+(B)$);
\end{tikzpicture}
\end{document}


this one uses the + syntax which means that the coordinate is realtive but without setting it as new reference point, as the ++ does.

If you are using this many times, you may like to create a macro. This is to illustrate how to convert Tobi's excellent code into one.

\documentclass{article}% ... or whatever you like

\usepackage{tikz}
\usetikzlibrary{arrows,calc}

\newcommand\crossprod[4][5]{%
\begin{tikzpicture}[
vec/.style={thick,[-)},
]
% Step 2
\coordinate (A) at #3;
\coordinate (B) at #4;
\coordinate (cross prod) at #2;
\def\tick{0.2}
% Step 3
\draw [-stealth] (-1,0) -- ($(#1,0)+(0.5,0)$) node [below] {$x$};
\draw [-stealth] (0,-1) -- ($(0,#1)+(0,0.5)$) node [left]  {$y$};
\foreach \x in {1,...,#1}
\draw (\x,\tick/2) -- ++(0,-\tick) node [below] {\x};
\foreach \y in {1,...,#1}
\draw (\tick/2,\y) -- ++(-\tick,0) node [left] {\y};
% Step 4
\draw [vec] (1,-1) -- ++(A) node [midway,below] {$A$};
\draw [vec] (-1,4) -- ++(B) node [midway,left]  {$B$};
% Step 5
\fill [gray!30] (cross prod) -- +(A) -- +($(A)+(B)$) -- +(B);
\draw [thick] ($(cross prod)+(A)$) -- (cross prod) -- ($(cross prod)+(B)$);
\draw [thick,dashed] ($(cross prod)+(A)$) -- ($(cross prod)+(A)+(B)$)
node [midway,right] {$A \times B$} -- ($(cross prod)+(B)$);
\end{tikzpicture}
}

%% #1 -- Optional argument for length of x and y axes; default is 5. Adjust for suitability
%% #2 -- upper left edge of the cross product. Adjust for suitability
%% #3 -- vector A as coordinate, starting from the origin.
%% #4 -- vector B as coordinate, starting from the origin.

\begin{document}
\crossprod{(1,5)}{(3,0)}{(0,-2)}

\crossprod[6]{(2,5)}{(3,-2)}{(-0.5,-2)}
\end{document}


This is a simple code:

\begin{tikzpicture}[font={\small}]
\draw[-latex] (-0.5,0) -- (5.5,0) node[below] {$x$};
\draw[-latex] (0,-0.5) -- (0,5.5) node[left] {$y$};
\foreach \x in {1,2,3,4,5}
{
\draw (0.1,\x) -- (-0.1,\x) node[left] {$\x$};
\draw (\x,0.1) -- (\x,-0.1) node[below] {$\x$};
}
\path[fill=lightgray] (1,2) rectangle +(3,2);
\draw[thick] (1,2) -- (1,4) -- (4,4);
\draw[thick,dashed] (1,2) -- (4,2) -- (4,4) node[right,midway] {$A\times B$};
\draw[very thick] (1,-1) node {\small [} -- (4,-1) node {)} node[below,midway] {$A$};
\draw[very thick] (-1,2) node[rotate=90] {[} -- (-1,4) node[rotate=90] {)}
node[left,midway] {$B$};
\end{tikzpicture}