How to make \cdot operator same width as / (division slash) operator and vice versa

You can use a \makebox to create a box as wide as the slash:

Code:

\documentclass{article}
\usepackage{amsmath}
\usepackage{calc}

\newcommand*{\mycdot}{\makebox[\widthof{/}]{$\cdot$}}
\newcommand*{\myslash}{\makebox[\widthof{${}\cdot{}$}]{{}/{}}}

\begin{document}\noindent
\verb|\cdot| same width as slash:
\begin{align*}
x &= a \mycdot a \\
x &= a / a
\end{align*}
Slash same width as \verb|\cdot|:
\begin{align*}
x &= a \cdot a \\
x &= a \myslash a
\end{align*}
\end{document}

This scales in subscripts and superscripts and allows for different widths based on other operators.

\documentclass{article}
\usepackage{amsmath}

\makeatletter
\newcommand\wslash[1][\cdot]{\mathbin{\mathpalette\w@slash{#1}}}
\newcommand\w@slash[2]{%
  \settowidth\dimen@{$\m@th#1#2$}%
  \makebox[\dimen@]{$\m@th#1/$}%
}
\makeatother

\begin{document}

\begin{align}
&a\cdot a\\
&a\wslash a \\
&a+a\\
&a\wslash[+]a\\
&x_{a\cdot a}\\
&x_{a\wslash a}
\end{align}

\end{document}