Just as a scalar can always be made up by adding any number of other scalars vectors can be made up by adding a suitable set of other vectors. If I insist on making up the vector by adding vectors which lie only along specific directions I can end up with a unique way of resolving the vector into its components along these specific directions. A sufficient condition is that the directions chosen are all perpendicular to one another and this is the type of situation which will be relevant here.
If we consider the two dimensional case I draw a vector and the two
unit vectors identifying the directions along which I wish
to resolve the vector. I can write the resolution as
Here I have chosen the typical notation used in this situation.
From Pythagoras we know that
Looking at the triangle formed by 
and 
I can also
write equations relating the angle between these vectors and 
,
and 
.
These equations are useful if we need to determine the components
of a vector with a known location with respect to the coordinate system.
If we need to go in the opposite direction we can use
Here it is however very important to realize that solving this equation
for theta does not give an unambiguous result. All I can say is that
To determine n I need to
figure out which of the two possible quadrants I am
actually in by looking at 
and individually.
Specifically if you punch Atan on your calculator you will always get
an angle between 
and 
indicating that the vector lies in the
first or fourth quadrant of the coordinate system. You need to look at
the signs of and individually to determine whether
is actually in the second rather than the fourth quadrant or in the third
rather than the first quadrant. Here is an example:
What is the magnitude and polar angle of the vector with coordinates
(-4,-2)?
The magnitude we get from Pythogaros :
Concerning the angle we can say that
Where I have decided to use degrees this time. Looking at the individual
components I decide that the vector lies in the third quadrant which means
that n=1 and
