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Usually the riddle is asked in the form "Why did the X cross the Y and the Z?", or "What do you get when you cross a Y with a Z?", the answer being as above (but more droll when, say, Y is a banana and Z is a gorilla).
If Y or Z is "a mountain climber", there's the rejoinder, "But a mountain climber is a scalar."
as I understand it, the Chicken in question would avoid getting to either point the vector is headed for if it crosses them both, wouldn't it? Or does it cross from one vector to the other? At which the answer would make perfect sense to me.