Why do we fly across the globe latitudinally (horizontally) instead of longitudinally?

Question

For example, if I were in Tangier, Morocco, and wanted to fly to Whangarei, New Zealand (the antipode on the globe) – wouldn’t it be about the same time to go up instead of across?

ETA: Thanks so much for the detailed explanations!

For those who are wondering why I picked Tangier/Whangarei, it was just a hypothetical! The-Minmus-Derp explained it perfectly: *Whangarei and Tangier airports are antipodes to the point that the runways OVERLAP in that way – if you stand on the right part if the Tangier runway, you are exactly opposite a part of the Whangarei runway, making it the farthest possible flight.*

In: 2270

Answer 1

Actually we typically fly by the shortest distance which AI be a “great circle” whose diameter plane cuts through the center of the earth. It may go over one of the poles.