Find the values of `x,y,z` if the matrix `A=[[0,2y,z],[x,y,-z],[x,-y,z]]` satisfy the equation `A^T A=I_3`

Explanation

`[[0, x, x], [2y, y, -y], [z, -z, z]][[0, 2y, z], [x, y, -z], [x, -y, z]]=[[1, 0, 0], [0, 1, 0], [0, 0, 1]]`

`[[0+x^2+x^2 ,0+xy-xy, 0-xz+xz], [0+xy-xy, 4y^2+y^2+y^2, 2zy-zy-zy], [0-xz+xz, 2zy-zy-zy, z^2+z^2+z^2]]=[[1, 0, 0], [0, 1, 0], [0, 0, 1]]`

`[[2x^2, 0, 0], [0, 6^y, 0], [0, 0, 3z^2]]=[[1, 0, 0], [0, 1, 0], [0, 0, 1]]`

`2x^2=1=>x=+-sqrt(1/2)`

`6y^2=1=>y=+-sqrt(1/6)`

`3z^2=1=>z=+-sqrt(1/3)`