How do I represent wind direction on a UTM (Universal Transverse Mercator) map projection?
Meteorological winds are referenced from North being 0$^{\circ}$ increasing in a clockwise direction.  Additionally, they are named for the direction the wind is coming from.  Your vector on the map, which is mathematically 45$^{\circ}$ is meteorologically 225$^{\circ}$ aka a southwesterly wind.


A 0$^{\circ}$ wind is a wind from the north (math vector 270$^{\circ}$).  
A 180$^{\circ}$ wind is wind from the south (math vector 90$^{\circ}$).
A 50$^{\circ}$ wind is from the northeast (math vector 220$^{\circ}$).


In general if $\alpha$ is the mathematical angle and $\beta$ is the meteorological angle, then the relation between the two conventions is given by $$\alpha = 270^{\circ} - \beta$$ where angles are specified in degrees.


Image from https://www.e-education.psu.edu/meteo300/node/719 from Penn State available under CC-by-nc-sa-3.0