A materials scientist is studying the symmetry of molecular arrangements in a self-healing material. The arrangement has a rotational symmetry group isomorphic to the dihedral group \(D_4\). How many distinct vectors \(\mathbfv = eginpmatrix x \ y \endpmatrix\) in \(\mathbbR^2\) with integer components satisfy \( \mathbfv \cdot \mathbfv = 10 \) and are invariant under a \(90^\circ\) rotation about the origin?