Entanglement witnesses (EWs) are a fundamental tool for detecting quantum entanglement in low-dimensional systems. We investigate the existence of product vectors in the range spaces of entanglement witnesses in \(3\times 3\) quantum systems. Through systematic analysis, we first prove that for EWs with inertia \((1,5,3)\) , there exists at least one product vector in their range spaces under specific kernel conditions. Furthermore, we establish a universal property that every entanglement witness \(W \in \mathcal {N}_{3,3}\) contains at least one nonzero product vector in its range space. As an application, our results provide new insights into the geometric structure of entanglement witnesses and may lead to novel methods for entanglement detection.