Answer
Yes, addition of vectors is cumulative.
Work Step by Step
Let's say that each leg of the original flight is represented by three vectors, u, v, and w.
The displacement from the original location is u+v+w after the three legs are counted.
Addition of vectors is inherently commutative, so for two vectors u,v, u+v = v+u.
In the same realm, u+v+w = u+w+v = v+w+u, etc...
So, regardless of what order the legs are in, the final position remains the same.
