You travel 3 m north in 1s, and 4 m east 2 s.

 

Displacement: 5 m in 3s. Average veolcity is 5/3 m/s.

 

 

or

 

V north = 3 m/s, and v east = 2 m/s. Average velocity for trip is (allegedly) 3.6 m/s. Obviously this wrong, but conceptually I'm not quite sure how to explain why it doesn't work.

You travel 3 m north in 1s, and 4 m east 2 s.

 

Displacement: 5 m in 3s. Average veolcity is 5/3 m/s.

 

 

or

 

V north = 3 m/s, and v east = 2 m/s. Average velocity for trip is (allegedly) 3.6 m/s. Obviously this wrong, but conceptually I'm not quite sure how to explain why it doesn't work.

 

 

For the latter, the duration of the trips is not the same.

For a start, you must distinguish between speed and velocity. Velocity is a vector and has one component of speed and the other of direction. Speed is just a scalar quantity. Are you asking about average speed (easy) or average velocity (not quite so easy)?