Question 755269
Think of your vector as the arrow from point O to point A:
{{{drawing(300,300,-2.5,2.5,-2.5,2.5,
grid(0),
blue(circle(0,0,0.05)),blue(circle(2,-sqrt(3),0.05)),
blue(arrow(0,0,2,-sqrt(3))),
locate(0,0.3,O)
locate(1.4,-1.8,A(2,-sqrt(3)))
)}}}
Point A is in the fourth quadrant.
It is at a distance {{{sqrt(7)}}} from O.
The coordinates of A are
{{{x=2=sqrt(7)*cos(theta)}}}
{{{y=-sqrt(3)=sqrt(7)*sin(theta)}}}
Your direction angle is
{{{theta=arctan(-sqrt(3)/2)=-40.9^o}}}
The angle {{{theta=arctan(y/x)}}} does not help you figure out the quadrant because <-2,sqrt(3)> would give you the same angle.
The signs of {{x}}} and {{{y}}} tell you the quadrant.
<(2,-sqrt(3)> and <(-2,sqrt(3)> are mirror images of each other.
<(-2,sqrt(3)> forms a {{{180^o-40.9^o=139.1^o}}} with the positive x-axis.