The vector (0, 0, 0) cannot be orthogonal to another vector because it has no direction.

in reality (0,0,0) is orthogonal to every vector, since the scalar product between him and any vectors is always 0

You are right. It depends how you define it.
The definition I use is that two vectors are orthogonal if they form a right angle.

Ok that's fine.
This concept is not at all trivial.
The physicists say that a vector is an object that has 3 features: length, direction and verse.
So if (0,0,0) is a vector must have one direction!!
But rightly as you say it has no direction.
On the other side the mathematicians define a vector as an object that "live" in a space with a certain number of dimensions: in this case 3.
So (0,0,0) is a vector respect to this point of view.
Moreover it's true that if two vector form a right angle, their scalar product is 0, and this is true not by definition but it is a fact. You can check it by using whatever method you prefer...arctg,Pythagorean theorem,...
So I think that the null vector is an extension of the physical vector concept, like I'm sure you know it happens for the elements of an Hilbert space on quantum physics, where the L2 complex functions are the vectors and the scalar product is an integral!!
That's really cool!!

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