We analyze the potential for using multiple transmission dimensions in a cognitive radio network (CRN) in terms of the probability of successful communications. We consider a random CRN where the users are randomly distributed in an area of interest. The users can be mobile. We derive the successful communication probability (SCP) with respect to transmit power for different density of primary users and secondary users by including different transmission dimensionalities such as time, frequency and antenna directionality. It is shown that using multiple transmission dimensions improves the SCP significantly. The advantage of using directional antennas for spatial reuse and increased range is also shown in terms of improved SCP.