Comment by bubblyworld

Sorry, on rereading this I've actually written a bunch of nonsense. It's true that each choice of n gives you back a metric, but the |x|, |y|, |z| actually refer to the components of your vectors. So in 1-dim space you just have |x|, in 2-dim space you have |x| and |y|, 3-dim space has |x|, |y| and |z| and so forth.

So in 3-dimensional space, for n=3, the "length" of a vector u=(u_x, u_y, u_z) is: d(u) = (|u_x|^3 + |u_y|^3 + |u_z|^3)^(1/3)

In 2-dimensional space, for n=4 and u=(u_x, u_y) you get the following: d(u) = (|u_x|^4 + |u_y|^4)^(1/4)

If you want to use the norm as a metric to work out the "distance" between two vectors u and v then you just compute d(u-v) where u-v is ordinary vector subtraction.