= Solution
For the particular <eigenvectors> chosen in (iii), the <dot products> are
$$
\mathbf e_1\cdot\mathbf e_2=0,\qquad
\mathbf e_1\cdot\mathbf e_3=2,\qquad
\mathbf e_2\cdot\mathbf e_3=-6.
$$
Thus \b[only $\mathbf e_1$ and $\mathbf e_2$ are <orthogonal>] for this choice. The <matrix> is not symmetric, so distinct <eigenvalues> alone do not imply <orthogonality>. Another basis within the repeated <eigenspace> may change which chosen <vectors> are perpendicular; the dot products above specify the convention completely.
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