Dot and Cross Products & Vector Identities

Snippets Page

Code for dot and cross product of three vectors.

Source Code

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

Additional Diagrams

Here is a special identity:

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

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

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