Primitive sets of vectors are very important in the theory of point lattices, since they constitute the sets of vectors that are part of a basis for the lattice.

A set of integer vectors $v_1,\ldots,v_k$ is primitive in $\mathbb{Z}^n$ if there are $v_{k+1},\ldots,v_{n}$ such that the matrix whose columns are $v_i$ has determinant 1 (i.e., unimodular). Is there an easy way to characterize primitive vectors which are orthogonal? In the end, I want to find families of vectors $v_1(m),\ldots,v_k(m)$ which are primitive, orthogonal and the norm increases with m. Is there any such family, is it known?