  Let $V$ be a finite dimensional vector space over a finite field. Suppose
that $\mathscr{F}_1$, $\mathscr{F}_2$, $\dots$, $\mathscr{F}_r$ are $r$-cross
$t$-intersecting families of $k$-subspaces of $V$. In this paper, we determine
the extremal structure when $\prod_{i=1}^r|\mathscr{F}_i|$ is maximum under the
condition that $\dim(\bigcap_{F\in\mathscr{F}_i}F)<t$ for each $i$.
