Final answer:
The patch of hair cells and supporting cells on the saccule's wall is called the macula sacculi. It contributes to the detection of head position and works with the vestibular system to maintain balance.
Step-by-step explanation:
The term that refers to the patch of hair cells and supporting cells positioned vertically on the wall of the saccule is the macula sacculi. The macula in both the utricle and saccule is composed of hair cells that are surrounded by supporting cells. These hair cells have stereocilia which extend into the viscous gel of the otolithic membrane. Above this membran is a layer of calcium carbonate crystals, called otoliths, which make the otolithic membrane top-heavy. Changes in head position lead to movements of this membrane, causing the stereocilia to bend and hair cells to depolarize or hyperpolarize. This information is interpreted by the brain to detect the position of the head in space, aiding in balance and spatial orientation.