One would also like to have a good definition of multiplication.
Well there are a couple of common ways of multiplying a vector by
a vector, but let's first ask how to multiply a vector such
as 
by a real number. For example what would be a good
definition of 
? Well you'd think it'd be good
if 
. I think it's good too, so lets
say that. Then by the procedure for addition discussed above,
we see that we're taking one vector and adding an identical
copy of it to itself, so the head of the first is at the tail
of the second (I hope these vectors don't smell). This gives a
vector going in the same direction but with twice the magnitude.
So in general that's what multiplication of a real number r by
a vector is going to do. It'll give us something going
in the same direction with r times its original magnitude.
In terms of components 
has x component 
and
y component 
.
If r equals -1, then the resultant vector points in the opposite
direction. For example if looks like 
, then
.