Dot and Cross Products & Vector Identities

Snippets Page

Code for dot and cross product of three vectors.

# Source Code

$$\bfu\times\bfv = \tikz{\node[vertex]at(0,.6){}edge(0,1)edge[bend right]node[vector,pos=1]{\bfu}(-.5,0)edge[bend left]node[vector,pos=1]{\bfv}(.5,0);} \quad\text{and}\quad \bfu\cdot\bfv = \tikz{\draw(0,0)node[vector]{\bfu}to[out=90,in=90,looseness=2](1,0)node[vector]{\bfv};}.$$


Here is a special identity:

    \tikz[scale=.6]{
\node[vertex](node)at(.5,1){}
edge[bend right]node[vector,pos=1]{$\bf u$}(0,0)
edge[bend left]node[vector,pos=1]{$\bf v$}(1,0);
\node[vertex](node2)at(2.5,1){}
edge[bend right]node[vector,pos=1]{$\bf w$}(2,0)
edge[bend left]node[vector,pos=1]{$\bf x$}(3,0)
edge[bend right=90](node);
}
=
\tikz[scale=.6]{
\draw(0,0)node[vector]{$\bf u$}to[bend left=90,looseness=2](2,0)node[vector]{$\bf w$};
\draw(1,0)node[vector]{$\bf v$}to[bend left=90,looseness=2](3,0)node[vector]{$\bf x$};
}
-\tikz[scale=.6]{
\draw(0,0)node[vector]{$\bf u$}to[bend left=90,looseness=1.7](3,0)node[vector]{$\bf x$};
\draw(1,0)node[vector]{$\bf v$}to[bend left=90,looseness=2.5](2,0)node[vector]{$\bf w$};
}


And the triple vector identity (also listed on its own page):

    \tikz[scale=.6]{
\node[vertex](node)at(.5,2){};
\draw[](0,0)node[vector]{$\bf u$}to[out=90,in=-135](node);
\draw[](1,0)node[vector]{$\bf v$}to[out=90,in=-45](node);
\draw[](2,0)node[vector]{$\bf w$}to(2,2);\draw(2,2)to[out=90,in=90,looseness=1.5](node);
}
=\tikz[scale=.6]{
\node[vertex](node)at(1.5,2){};
\draw[](0,0)node[vector]{$\bf u$}to(0,2);\draw(0,2)to[out=90,in=90,looseness=1.5](node);
\draw[](1,0)node[vector]{$\bf v$}to[out=90,in=-135](node);
\draw[](2,0)node[vector]{$\bf w$}to[out=90,in=-45](node);
}
=\tikz[scale=.6]{
\node[vertex](node)at(1,2.5){};
\draw[](0,0)node[vector]{$\bf u$}to[out=90,in=-90](1.5,1.75);\draw(1.5,1.75)to[out=90,in=-45](node);
\draw[](1,0)node[vector]{$\bf v$}to[out=90,in=-90](2.5,2)to(2.5,2)to[out=90,in=90,looseness=2](node);
\draw(2,0)node[vector]{$\bf w$}to[out=90,in=-90](.5,1.75);\draw[](.5,1.75)to[out=90,in=-135](node);
}
=\tikz[scale=.7]{
\node[vertex](node)at(1,2){};
\draw[](0,0)node[vector]{$\bf u$}to[out=90,in=-135](node);
\draw[](1,0)node[vector]{$\bf v$}to[out=90,in=-90](node);
\draw[](2,0)node[vector]{$\bf w$}to[out=90,in=-45](node);
}


And finally another triple vector identity:

    \tikz[scale=.6]{
\node[vertex](node)at(.5,1.5){};\node[vertex](node2)at(1,2.5){};
\draw[](0,0)node[vector]{$\bf u$}to[out=90,in=-135](node);
\draw[](1,0)node[vector]{$\bf v$}to[out=90,in=-45](node);
\draw(node)to[out=90,in=-135](node2);
\draw[](2,0)node[vector]{$\bf w$}to[out=90,in=-45](node2);
\draw(node2)to[out=90,in=-90](1,3.25);
}
=\tikz[scale=.6]{
\draw[](0,0)node[vector]{$\bf u$}to[out=90,in=90,looseness=2](2,0)node[vector]{$\bf w$};
\draw[](1,0)node[vector]{$\bf v$}to[out=90,in=-90](1,3.25);
}
-\tikz[scale=.6]{
\draw[](0,0)node[vector]{$\bf u$}to[out=90,in=-90](1,3.25);
\draw[](1,0)node[vector]{$\bf v$}to[out=90,in=90,looseness=2.5](2,0)node[vector]{$\bf w$};
}

page revision: 6, last edited: 15 Feb 2010 15:44