The alignment of Large Language Models (LLMs) for multi-turn conversations typically relies on reward signals derived from the content of the text. This approach, however, overlooks a rich, complementary source of signal: the dynamics of the interaction itself. This paper introduces TRACE (Trajectory-based Reward for Agent Collaboration Estimation), a novel reward signal derived from the geometric properties of a dialogue's embedding trajectory--a concept we term'conversational geometry'. Our central finding is that a reward model trained only on these structural signals achieves a pairwise accuracy (68.20%) comparable to a powerful LLM baseline that analyzes the full transcript (70.04%). Furthermore, a hybrid model combining interaction dynamics with textual analysis achieves the highest performance (80.17%), demonstrating their complementary nature. This work provides strong evidence that for interactive settings, how an agent communicates is as powerful a predictor of success as what it says, offering a new, privacy-preserving framework that not only aligns agents but also serves as a diagnostic tool for understanding the distinct interaction patterns that drive successful collaboration.