mA = {{7, -4, -2}, {-4, 1, -4}, {-2, -4, 
          7}}

Eigensystem[mA]

mQ = 
      Transpose[{(1/(Sqrt[2])) {-1, 0, 1}, (1/(Sqrt[3])) {-1, 
              1, -1}, (1/(Sqrt[6])) {1, 2, 1}}]

Simplify[Transpose[mQ] . mA . mQ]

vx1 = mQ . {1, 0, 0}

vx2 = mQ . {0, 1, 0}

vx3 = mQ . {0, 0, 1}

Simplify[vx1 . (mA . vx1)]

Simplify[(Cos[t] vx1 + 
            Sin[t] vx2) . (mA . (Cos[t] vx1 + 
                Sin[t] vx2))]

Show[
Graphics3D[{

{Black, Thickness[0.01], 
            Line[Table[ 
                Cos[t] vx1 + Sin[t] vx2, {t, 0, 2 Pi, 
                  (Pi/64)}]]}, 
{PointSize[0.02], 
            GrayLevel[0.5], 
            Table[Point[Cos[t] vx1 + Sin[t] vx2], {t, 0, 2 Pi, 
                (Pi/32)}]}, 
{Thickness[0.005], 
            GrayLevel[0.5], 
            Table[Line[{{0, 0, 0}, Cos[t] vx1 + Sin[t] vx2}], {t, 0, 2 Pi, 
                (Pi/16)}]}, 
{Thickness[0.01], Cyan, 
            Line[{{0, 0, 0}, vx1}], Magenta, Line[{{0, 0, 0}, vx2}], Yellow, 
            Line[{{0, 0, 0}, vx3}]}, 
{Thickness[0.01], 
            Red, Line[{{0, 0, 0}, {1, 0, 0}}], Green, 
            Line[{{0, 0, 0}, {0, 1, 0}}], Blue, 
            Line[{{0, 0, 0}, {0, 0, 1}}]}
}], 
      Axes -> True, 
      AxesEdge -> {{1, -1}, {1, -1}, {-1, 1}}, 
      AxesLabel -> {x1, x2, x3}, 
      Ticks -> {{-1, 0, 1}, {-1, 0, 1}, {-1, 0, 1}}, 
      ViewPoint -> {2.5445336142692656`, 0.9334761761768039`, 
          0.4308142457766752`}, 
      PlotRange -> {{-1.1, 1.1}, {-1.1, 1.1}, {-1.1, 
            1.1}}
]

In the above plot the coordinate vectors are represented as red, green and blue lines. The unit eigenvectors of the given matrix are represented by cyan, magenta and yellow lines. Cyan and magenta correspond to the eigenvalue 9 and yellow corresponds to the eigenvalue -3.