Sometimes it'll be convenient to deal with vectors that
always have a magnitude of 1 but can have any direction.
Such vectors are called ``unit vectors''. So take again
our often used vector 
. If we want to make it
into a unit vector, we have to construct something with
the same direction but a magnitude of 1. We typically
but silly little party hats on the top of unit vectors
to make it clear to everyone that these are unit vectors.
How the party hat notation got started beats me, but
it does work out pretty nicely. Anyway getting back to
the problem at hand, we'd call this unit vector 
.
To actually write down a formula for in terms
of A isn't too tough. Just write
The right hand side means the vector multiplied
by the real number 
.
Let's check that this is correct. First of all, does it have
the right magnitude? From the definition of
multiplication above, when we multiply
the vector by a real number number, in this case
, the resulting vector has magnitude of
. That's correct.
How about the direction? That's right too, because when
you multiply a vector by any nonzero number, it also points
in the same direction as the original vector.
So this indeed is the right expression for .