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I think moments are the most useful tool here. Take two moments, that of the outline you have in yellow (just the outline) and the image inside the box that contains the outline. The central moments of the outline is the geometric center of the arrow, more or less. The central moments of the image around the outline is concentrated (because things besides black is what it's measuring) at the tail of the arrow.
Draw a line from the tail point you found through the center point you found, and there you are, a direction.
No promises though. Your edges are really fuzzy, which is throwing off the contours. Think you can fix that?
