# Dot products of three or more vectors

Can't we construct a mapping from $V^3(R^1)$ to $R$ such that $a.b.c = a_{x}b_{x}c_{x}+a_{y}b_{y}c_{y}+a_{z}b_{z}c_{z}$ (a,b,c are vectors in $V^3(R^1)$ ) and more generally $a^n$ , $a.b.c.d.e...$ mappings so that $(a^p)^{1/p}$ is the $p$ norm of vectors in $V^3$ ?

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It would be nice to have a nice way to write products like these. Similarly to $a^Tb$ for `dual' inner products. –  Thomas Ahle Mar 3 '14 at 21:28

I found the solution. To 'dot' together vectors $a$, $b$, $c$, make the diagonal matrix $B$ and write:
$a^T B c$