“Explore the Vault” is an objective in the mission Paradox Daily Heroic in Destiny: The Taken King
Head over to the pit and drop down into the next drop. This is a long one. Hunters, use the ledges to lower yourself down carefully. Titans and Warlocks can be a bit more reckless with their Glide or Lift abilities. Still, use the ledges to keep your speed under control. This drop is fatal
Head forward when you get to the bottom. Time to deal with another drop. Once again, control your descent. Just head forward and make your way to the Gorgons' Labyrinth. Just keep going forward until you reach the Labyrinth itself.
Move forward in the Labyrinth. You will see some Taken Hobgoblins scattered throughout the walls. Clear them out and then find the Taken Hydra