Yea, its my mistake, sorry. I was faster at typing than thinking. I thought they were orthogonal but obviously they cant be if two of them are (1, 0, 0), (0, 1, 0) and the third is rotated.

Here it is again complete, right (0.5, 0, -1), up (0, 1, 0), forward (1, 0, 0.5). Here they are orthogonal but if we normalize they become right (0.44, 0, -0.89), up (0, 1, 0), forward (0.89, 0, 0.44). Now they are also orthonormal.

The way Paradigm Shifter suggested it we could make a matrix from them like:

(044, 0, -0.89)

(0, 1, 0)

(0.89, 0, 0.44)

And the vector v is (-0.5, 0, -1). So I multiplied this matrix (lets call it M) with v and I got v' (0.67, 0, -0.88).

Now I wonder, are this the coords of a same point in space but in respect to the space of M ?

I really am not sure, always when I wanted to rotate some vector I would do it this way. So I think that this vector v' is a rotated vector and not the same vector in space.