Difference between metric and norm made concrete: The case of Euclid This is a follow-up question on this one. The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me.
This time I am making things more concrete: I am esp. interested in the difference between a metric and a norm. I understand that the metric gives the distance between two points as a real number. The norm gives the length of a a vector as a real number (see def. e.g. here). I further understand that all normed spaces are metric spaces (for a norm induces a metric) but not the other way around (please correct me if I am wrong).
Here I am only talking about vector spaces. As an example lets talk about Euclidean distance and Euclidean norm. Wikipedia says:

A vector can be described as a
  directed line segment from the origin
  of the Euclidean space (vector tail),
  to a point in that space (vector tip).
  If we consider that its length is
  actually the distance from its tail to
  its tip, it becomes clear that the
  Euclidean norm of a vector is just a
  special case of Euclidean distance:
  the Euclidean distance between its
  tail and its tip.

What confuses me is that they seem to be having it backwards: The Euclidean metric induces the Euclidean norm: You measure the distance between tip and tail and get the length out of that. What makes my confusion complete is that $L^2$ distance is also called the Euclidean norm (see here).
I would very much appreciate it if somebody could clear the haze.
 A: The simplest answer to the question in the title is that a metric is a function of two variables and a norm is a function of one variable. 
The other question, which I would summarize as "which came first?" is (at least in the Euclidean context) a chicken-and-egg question. You can define the Euclidean (or $L^2$) distance between $x$ and $y$ as $\sqrt{(x_1-y_1)^2+\cdots+(x_n-y_n)^2}$ and then define the norm as the distance from $x$ to the origin; or, you can define the Euclidean (or $L^2$) norm as $\sqrt{x_1^2+\cdots+x_n^2}$ and then define the distance from $x$ to $y$ as the norm of $x-y$. 
A: The metric $d(u,v)$ induced by a vector space norm has additional properties that are not true of general metrics. These are:
Translation Invariance: $d(u+w,v+w)=d(u,v)$
Scaling Property: For any real number $t$, $d(tu,tv)=|t|d(u,v)$.
Conversely, if a metric has the above properties, then $d(u,0)$ is a norm.
More informally, the metric induced by a norm "plays nicely" with the vector space structure.  The usual metric on $\mathbb{R}^n$ has the two properties mentioned above.  But there are metrics on $\mathbb{R}^n$ that are topologically equivalent to the usual metric, but not translation invariant, and so are not induced by a norm.
