Pictorially vector addition is pretty straightforward.
If you want to add the vectors 
and 
, you
just take the second vector, represented by an arrow, and
translate it so that it's tail is at the head of the
first vector. 
is just a vector that starts at
the tail of the first arrow and goes to the head of the
second, so the whole thing, vectors , , and
, form a triangle, as shown:
Click here to see full figure
Now the nice thing about this definition is that it's
easily seen that 
, and also the rule
for adding in terms of components couldn't be simpler.
Look at the x components for example. What is the
x component of the vector 
?. Its just the
addition of the x components of the two vectors,
. This is easily seen by the following
picture. We just show ``project'' the components
of A, B, and A+B down close to the x - axis. You
see there's a of length 
, one of length 
,
and one which is the x component of which
I labeled 
.
You see from that picture that the total length of
the red and green lines, is the length of the blue line.
That's what we wanted to show. The same holds true
in the y direction, so that the y component of
is just 
.
So vectors add nicely in terms of their x and y components. It's much messier in terms of the vectors polar representation.