Multi align equation right

\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage[latin1]{inputenc}% really? that is very 20th century, why not utf8?

\begin{document}


\begin{alignat*}{3}
  r(\theta)    ={}&&  (x_0 + a \cos x) i  + {}&& (y_0 + a \sin x) j \\
  r'(\theta)   ={}&&      - a \sin x  i   + {}&&       (a \cos x) j \\
  F(r(\theta)) ={}&&                0 i   + {}&& (x_0 + a \sin x) j
\end{alignat*}
\end{document}

alignat is a variant of align that avoids adding extra space. You need && so you only use the right aligned columns not the left aligned, and then you need {} after the = and + to preserve the correct spacing as they are before the & not after.