In this paper, we study the position vectors of a spacelike W-curve (or a helix), i.e., curve with constant curvatures, with spacelike, timelike and null principal normal in the Minkowski 3-space $mathbb{E}_1^3$. We give some characterizations for spacelike W - curves whose image lies on the pseudohyperbolical space $mathbb{H}_0^2$ and Lorentzian sphere $mathbb{S}_1^2$ by using the positions vectors of the curve.