You can also use polar coordinates to represent a two dimensional vector. You just specify the angle the vector makes with respect to the x axis. In the case of the first coordinate system we used, the angle is 56 degrees as shown:
Click here to see full figure
The the x and y components of the vector are also shown.
From trigonometry we can relate the angle of 56 degrees,
(call it theta for more generality) to the x and y
components, 
, and 
:
Then you have to talk about the length or ``magnitude''
of the vector.
The magnitude of the vector can be denoted by the absolute
value symbol 
.
Sometimes one doesn't want to write it as
because it looks too complicated, instead one can write it
as A (not bold face any more). If you are writing it by
hand, instead of writing 
you'd also write A.
So from good old Pythagoras we have
In three dimensions this is just 
.
Sometimes this polar representation is a more useful way
of understanding a final answer then by looking at 
.
You know immediately the magnitude of the vector, and
the direction it's pointing in. But using x and y components
tends to be the most useful way of wading through intermediate
steps of a problem. We'll see that operations performed on
vectors are easiest done in terms of these components.
We can go the other way and relate the x and y components of
to A and 
. Again it's just trigonometry:
