Difference between angle and direction in two dimensional space? Is there any difference between angle and direction in relation to 2D space? 
Can one of them help me in a way that the other can not?
the only difference that I can see is that angle is single value and the other is a vector. 
Thank you in advance.
 A: Depends on what exactly you mean by "direction".
One way to "describe" the word direction is by sayin that it's the "property" that $x$ shares with $\lambda x$ for all $\lambda>0$. In this case, you can "represent" the direction of a nonzero vector $x$ by the normalized vector $\frac{x}{\|x\|}$, and the set of all directions is the set $S^1$ (i.e., the unit circle).
This set is in a 1-to-1 correspondence with the set of all angles, as you can map any angle $\theta$ to the point $(\cos\theta, \sin\theta)$. 

So, in this way, directions and angles, in $2D$, are two sides of the same coin -- if I tell you the angle between a vector and the positive $x$-axis, I give you the exact same information as I would if I told you what the normalized version of that vector is.
A: From an axiomatic point of view one can say that angle is a numerical measurement of direction. 
In a formal treatment of Euclid's axioms, for example, "direction" is defined first (perhaps one might say that "direction" and "ray" are equivalent), and "angle" is then introduced as a numerical measurement assigned to a pair of directions: the angle between two directions represents intuitively the amount one must rotate one direction to get to the other.
Next, when a Cartesian $x,y$-coordinate system is imposed on a bare Euclidean plane, each individual direction may be assigned an angle, equal to the angle between that direction and the direction of the positive $x$-axis.
