I can show that they are orthogonal and also find the orthonormal basis by the Gram-Schmid method. But I don't understand how the coordinate systems axes are defined by them? Could anyone point me in the right direction? Also please feel free to tell me if I worked it out wrong.
$\vec{X} = (1,1,1),\vec{Y} = (-1,2-1), \vec{Z}=(-3,0,3)$
So $\vec{X}\cdot\vec{Y} = 0, \vec{Z}\cdot\vec{X} = 0, \vec{Z}\cdot\vec{Y} = 0$, So they are orthogonal.
Basis B;
$||\vec{X}|| = \sqrt1^2+1^2+1^2 = \sqrt3$
$||\vec{Y}|| = \sqrt-1^2+2^2+-1^2 = \sqrt2$
$||\vec{Z}|| = \sqrt-3^2+0^2+3^2 = 0$
$B = span(\frac{1}{\sqrt3},\frac{1}{\sqrt3},\frac{1}{\sqrt3}),(\frac{-1}{\sqrt2},\frac{2}{\sqrt2},\frac{-1}{\sqrt2}), (-3,0,3)$