Place the yogurt and sour cream in a medium-sized mixing bowl.
Peel cucumber and cut in half lengthwise. Remove the seeds by scraping them out with a spoon. Grate one half of the cucumber until you have 1/2 cup grated cucumber. Reserve the remaining cucumber. Place the grated cucumber in the mixing bowl with the yogurt and sour cream.
Add lemon juice, garlic powder, dried dill weed, salt, and black powder. Stir until the ingredients are evenly mixed.
Cover the dip and place in the refrigerator to chill for at least 1 hour. Stir again just before using.
Cut the remaining cucumber into 1/4-inch slices. Arrange the cucumber slices, baby carrots, and broccoli florets on a platter. Serve with dip!