Solution
= Solution
For either sign, the <dot product> with $v_1$ is
$$
v_1\cdot v_\pm=(-5\pm\sqrt{15})+(2\mp\sqrt{15})+3=0.
$$
The remaining pair also satisfies
$$
v_+\cdot v_-=(25-15)+(4-15)+1=\boxed{0}.
$$
Hence the three <eigenvectors> are mutually <orthogonal>. This also follows from the <spectral theorem for real symmetric matrices>, since $M=M^T$ and its <eigenvalues> are distinct, but the displayed products verify the actual chosen <vectors> directly.