What is the Direction of a Zero (Null) Vector? To be more precise, I am interested in knowing if the intuition that a Euclidean zero vector does not have a particular direction is actually correct, and if there is a rigorous formulation that would back it up.

Wikipedia's entries on the zero vector seems to agree with that intuition, but as always, one shouldn't blindly trust Wikipedia: in one place, it is stated that a zero vector "is orthogonal to all other vectors with the same number of components," while in another, it is stated that "two vectors can be considered orthogonal if and only if their dot product is zero, and they have non-null length." Correct me if I am wrong, but these two statements contradict each other.

This question popped up in my head when I heard someone arguing that since a Euclidean vector is defined as a geometric entity that has both a magnitude and direction, and since a zero vector is a vector with length 0, then it is only fitting for a zero vector to have a "direction 0." I am personally inclined to say that a zero vector does not have a particular direction, but as I said, I would like to know if there is a rigorous formulation that would lead to this conclusion.

And to put the question in a less "discussion-inducing" form, is there an agreement on the direction of a Euclidean zero vector?
The zero vector has no particular direction; this is consistent with the fact that it is orthogonal to every other vector.  (It doesn't really make sense to say it has "direction 0", since direction is not a magnitude; "direction 0" makes no more sense than "direction 1"
or "direction 5.873".)   

Alternatively, you could say that it points in every direction, but with zero magnitude,
since if you take any vector and multiply it by zero, you get the zero vector.  "Every
direction" is the same as "no particular direction"; it's just a different way of phrasing
things.

This is one flaw with the traditional description of a vector as being a pair consisting of
a magnitude and a direction:  For the zero vector, the magnitude is zero, but the direction
is arbitrary.
I think it is possible for a zero magnitude vector to have direction.  Consider a particle moving in a circle at constant speed. Velocity is variable because direction changes.  magnitude of acceleration is zero because speed is constant. Acceleration (directed towards center) is variable because direction changes every instant. Clearly acceleration is present with magnitude zero.