A 38-year-old woman from Mumbai undergoes surgery for a right parotid pleomorphic adenoma. During dissection, the surgeon carefully identifies the external carotid artery and facial nerve to plan the superficial parotidectomy. The surgical team notes that the external carotid artery divides into its terminal branches within the parotid gland. Which of the following correctly describes the relationship between the external carotid artery and the facial nerve within the parotid gland?