[Linear Algebra] Spectral Theorem probably

This is one of those things where I though I knew what was going on, but I can't make the problem on the topic shake out-

For matrix

A=[(13; -4; -2); (-4; 13; 2); (-2; 2; 10)], find an orthogonal matrix T such that (T^t)AT is a diagonal matrix.

Since A is symmetric and real, it has an eigenvalue. Since A is symmetric and has an eigenvalue there exists a orthogonal matrix T and a diagonal matrix D such that (T^t)AT=D, and lastly, eigenvectors belonging to different eigenvalues are orthogonal to each other.

So - A has two eigenvalues, λ1=9 (with algebraic multiplicity 2) and λ2=18. A is symmetric, so the eigenspace associated with eigenvalue 9 has geometric multiplicity 2.

The eigenvectors associated with λ=9 are v_1=(1,1,0) and v_2=(1,0,2) respectively, applying gram-schmidt to orthogonalise gives

u_1=(1,1,0) and

u_2=(1,-1,4),

and lastly, vectors associated with separate eigenvalues are orthogonal, so the unit normal vector of the plane 2x_1-2x_2-x_3=0 u***_***3=(2,-2,-1) must be an eigenvector associated with λ2=18,

normalising and assembing in a suitable order to match the eigenvalues on the diagonal of D SHOULD, by my understanding, do it, since u_1-3 are orthogonal, and normalised T^(-1)=T^t and all that jazz.

My issue is that the given answer is

T=(1/3)[(1; 2; -2); (2; 1; 2); (-2; 2; 1)]

and believe you me, I've tried to figure out where the heck that matrix is coming from, which method you'd apply to get there, and I cannot for the life of me. What have I missed?