Why is $(0,0,0)$ not acceptable as a co-ordinate on the projective plane? Reading "Elliptic Tales: Curves, Counting, and Number Theory" which states $(0,0,0)$ cannot be a co-ordinate on the projective plane but I find the argument advanced for this in the book to be excessively hand-wavy (though I am not a mathematician).
Could someone supply an argument for this - and sorry to add the condition - but it probably needs to match the level of knowledge required for the book ("freshman" student or thereabouts).
 A: There are already some relevant aspects to this included in comments to your question. The key point here is that a point is not represented by a single vector, but by an equivalence class of vectors, namely all scalar multiples of a given vector. If you were to include the null vector into these classes, then by transitivity of the equivalence relation, all classes would become equal. This is the essence of Steven's comment. If, on the other hand, you were to define the null vector as an equivalence class all by itself, then that class would be fundamentally different from all other classes, since it would only contain a single element. Using the same term “point” for that special case does not seem justified, algebraically or geometrically, but more on that in my last paragraph.
You can also interpret points as one-dimensional subspaces of $\mathbb R^3$. Any such subspace can be described as the span of any vector it contains, except one: the null vector does not uniquely define such a one-dimensional subspace, since it is an element of every subspace, and its span has dimension zero. This is essentially what Ansgar's comment is about.
Yet another, very visual way to think of this: a point in the projective plane is pretty much like a direction. An arrow of any length can be used to describe a direction, and the length does not matter. But an arrow of length zero does not define a direction in any reasonable way, so it can not describe a point. (This analogy disregards the fact that for arrows and directions, orientation usualy does matter, but for vectors representing points in a projective plane, the sign does not matter.)
The null vector does occur in practice. E.g. if you compute the point of intersection between two lines, and the two lines happen to be one and the same, then the result is the null vector. But the result is geometrically best interpreted as “something undefined, ambiguous or degenerate happend during computation”. If you were to call that null vector a point nevertheless, then you would simplify some aspects of some axioms a bit, but complicate others quite a lot. So instead of “two different lines meet in a unique point of intersection”, you'd get something like “any two lines meet in a point of intersection, and also in the null-point, and the point of intersection is different from the null point if the lines are different and neither line is the null-line”. Not really useful.
