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Yes and no.

Some use "displacement" only to describe a change in the position of an object from some initial starting point to some ending point. That is, there is a distinction between "position" and "displacement." The position would be defined relative to a reference point. In that case the arrow depends only on where the particle was and where it ended up and the reference point does not matter.

Others use a definition for displacement which describes the difference between an object's position and a fixed reference point. That is, how far an object is displaced from a certain point even if the object had never been at that point. A distinction between position and displacement is not made. For this latter definition, the choice of reference point will make a difference in the direction of the arrow.

In physics problems one usually only cares about the changes in position (and velocity, etc) and the choice of reference point will not affect these.

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