>Mathematical vectors are a very general thing, they are just pairs,
>triples, tuples of something. In physics they are used that way,
>but in many cases vectors are homogeneous in physics. It�s very handy,
  To follow the weather map example, you could have wind velocity
vectors, each with three components:  the speed E-W, N-S and
up-down.  But you might alternatively choose to have vectors with
5 components:  the three speed components and also the temperature
and humidity at that point.  Programming-wise, it just a question
of what's more convenient.  However, while it makes sense to talk
about total wind speed as sqrt(vx**2+vy**2+vz**2), the quantity
sqrt(vx**2+vy**2+vz**2+temperature**2+humidity**2) is surely a
meaningless, useless number.

> It has four real components, but a different metric etc. (the
> fourth component behaves a little imaginary).
  The "different metric" is key.  While the space-distance is
sqrt(dx**2+dy**2+dz**2), the relativistic "distance" includes not
"+t**2" but rather "-(ct)**2".  But you can use regular
componentwise addition and subtraction.  Say baby Fred was born in
a hospital.  Baby Sam was born 20 feet to the east, 1 floor (10
feet) higher up, and 3 days later.  Baby Charles was born 10 feet
to the west of Sam, on the same floor, and 2 days earlier.
Ordinary, componentwise arithmetic tells us Charles was born 10
feet to the East of Fred, 1 floor up, and 1 day later.  But to
calculate the relativistic "distance" between the births, you need
to treat that time component differently.  So if your program was
using a generic package that does ordinary arithmetic on vectors
with N components, you would have to overide its ordinary
"distance" function with the relativistic one.