Ok,

Left Handed coordinate system mean "X" positive point from left to right, example:

(10,0,0) -----------> (20,0,0)

for to draw a triangle in front of your point of view you have to define in clockwise order.


Righ Handed coordinate system mean "X" positive point from Right to left,
example:

(20,0,0) <----------- (10,0,0)

for to draw a triangle in front of your point of view you have to define in anti-clockwise order.

When i define my vertices in clockwise order (from my point of view),
the resulting vector alway's points away from me.

Is this correct??
When you define your vertices in clockwise order and if your API graphics system is setuped in LEFT handed mode then you can see the triangle; and the normal vector point toward to your position.

When you define your vertices in clockwise order and if your API graphics system is setuped in RIGHT handed mode then you dont see the triangle (you are seeing the back which normally is culled); and the normal vector point away from your position.

The most commun use for normals is for lighting and for doing culling, becouse it tells if any desired point in the world (a light, a camera etc) is in front or back or inside the triangle.