Dot and Cross Products & Vector Identities

Snippets Page

Code for dot and cross product of three vectors.

Source Code

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

Additional Diagrams

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]{$\bf w$};
    }