  We give conditions on a general family $P_{\lambda}:\R^n\to\R^m, \lambda \in
\Lambda,$ of orthogonal projections which guarantee that the Hausdorff
dimension formula $\dim A\cap P_{\lambda}^{-1}\{u\}=s-m$ holds generically for
measurable sets $A\subset\Rn$ with positive and finite $s$-dimensional
Hausdorff measure, $s>m$, and with positive lower density. As an application we
prove for measurable sets $A,B\subset\Rn$ with positive $s$- and
$t$-dimensional measures, and with positive lower density that if $s + (n-1)t/n
> n$, then $\dim A\cap (g(B)+z) = s+t - n$ for almost all rotations $g$ and for
positively many $z\in\Rn$.
