  We prove effective joint equidistribution of several natural parameters
associated to primitive vectors in $\mathbb{Z}^{n}$, as the norm of these
vectors tends to infinity. These parameters include the direction, the
orthogonal lattice, and the length of the shortest solution to the associated
$\gcd$ equation. We show that the first two parameters equidistribute w.r.t.
the Haar measure on the corresponding spaces, which are the unit sphere and the
space of unimodular rank $n-1$ lattices in $\mathbb{R}^{n}$ respectively. The
main novelty is the equidistribution of the shortest solutions to the $\gcd$
equations: we show that, when normalized by the covering radius of the
orthogonal lattice, the lengths of these solutions equidistribute in the
interval $\left[0,1\right]$ w.r.t. a measure that is Lebesgue only when $n=2$,
and non-Lebesgue otherwise. These equidistribution results are deduced from
effectively counting lattice points in domains which are defined w.r.t. a
generalization of the Iwasawa decomposition in simple algebraic Lie groups,
where we apply a method due to A. Gorodnik and A. Nevo.
