Cutting a sandwich 'along the middle' is still considered a vertical cut, just along a different plane. Same as cutting corner to corner. Vertical but on a plane that isn't orthogonal to the major XYZ axes.
Lemonade is squeezed lemon and sugar, and maybe with something else like Strawberry, but fairly tart.
I don't know about American lemonade being tart. Every lemonade recipe I've tried has a fuckton of sugar added to it. Even when I cut the amount by 1/3 it's still super sweet.
That's just a 3d perspective along the Z axis