Question

In any problems involving circular motion, which way does the tangential speed vector point?

Answer 1

In what may be one of the most remarkable coincidences in
all of physical science, the tangential component of circular
motion points along the tangent to the circle at every point. 

The object on a circular path is moving in that exact direction
at the instant when it is located at that point in the circle.  The
centripetal force ... pointing toward the center of the circle ...
is the force that bends the path of the object away from a straight
line, toward the next point on the circle.  If the centripetal force
were to suddenly disappear, the object would continue moving
from that point in a straight line, along the tangent and away from
the circle.