Unlike many complex networks studied in the literature, social networks rarely exhibit regular unanimous behavior, or consensus of opinions. This requires a development of mathematical models that are sufficiently simple to be examined and capture, at the same time, the complex behavior of real social groups, where opinions and actions related to them may form clusters of different size. One such model, proposed in [1], deals with scalar opinions and extends the idea in [2] of iterative pooling to take into account the actors' prejudices, caused by some exogenous factors and leading to disagreement in the final opinions. In this paper, we offer a novel multidimensional extension, which represents the dynamics of agents' opinions on several topics, and those topic-specific opinions are interdependent. As soon as opinions on several topics are affected simultaneously by the same influence network, they automatically become related. However, we consider interdependent topics, and hence the opinions being formed on these topics are also mutually dependent. We examine rigorous convergence properties of the proposed model and find explicitly the steady opinions of the agents. Although our model assumes synchronous communication among the agents, we show that the same final opinion may be reached "on average" via asynchronous gossip-based protocols.