Is the squared euclidean norm a measure for the distance of two points?

Question

I like to prove that a measure for the distance $d$ of two points $\vec a$ and $\vec b$ in $R^N$ is given by the squared euclidean norm

$$d^2= \sum^N_j (a_j - b_j)^2 $$

So far I was able to show that the squared euclidean norm is no norm itself (triangular inequality not fulfilled, e.g. for $N=3$ and $\vec a =(4,4,4)$, $\vec b =(2,2,2)$, $\vec c =(0,0,0)$).

There should be a way to show this by using the minimization of a quadratic form but I can't find it. Any suggestions?

Edit:

Quadratic form as generalized distance covers the same topic...

Answer 1

No. Don't satisfies Triangle inequality.

For example

d([0 0 0],[2 2 2])=12 > d([0 0 0], [1 1 1]) + d( [1 1 1], [2 2 2])=6.