Direction of Resultant velocity So we know that an object undergoing horizontal circular motion, has a angular velocity with direction perpendicular to the plane of motion, and also a linear velocity tangential to the circular path.
Question : if we add linear and angular velocity,  wouldn't there be a resultant velocity, at an angle above the horizontal?
Why does the object still undergoes motion along the horizontal circle?
 A: You meant the angular velocity vector $\vec{\omega}$, I think. But adding the velocity vector $\vec{v}$ to the angular velocity vector $\vec{\omega}$ would be like adding apples to oranges.
Look even at the dimensions of the scalars of these vectors, for velocity that is $\mathrm{m/s}$, for angular momentum it is $\mathrm{s^{-1}}$ (angles have no dimension in the S.I. system).
Now look at this vectorial derivation of the centripetal acceleration vector: here the vectors that concern you are used in a vector product, resulting in the centripetal acceleration needed to keep an object in orbit, or horizontal circular motion as you preferred to call it.
A: You cannot add angular velocity to horizontal velocity...they are two diff things
A: in both cases the direction of vectors take 
if one vector which in angular velocity
and take another vector which in linear velocity
in that cases first perform rotational motion and another perform linear motion so resultant vector have no perfect direction.
