Find the eigenvalues and eigenvectors of the matrix $$\left[\begin{array}{ccc} 6 &-4 &-14\cr -6 &4 &14\cr 3 &-4 &-11 \end{array}\right].$$ From smallest to largest, the eigenvalues are $\lambda_1 < \lambda_2 < \lambda_3$ where

$\lambda_1 =$ has an eigenvector $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$,
$\lambda_2 =$ has an eigenvector $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$,
$\lambda_3 =$ has an eigenvector $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$.

Note: you may want to use a graphing calculator to estimate the roots of the polynomial which defines the eigenvalues.