Express \(x ^ { 2 } - 8 x + 11\) in the form \(( x - a ) ^ { 2 } + b\) where \(a\) and \(b\) are constants.
Hence write down the minimum value of \(x ^ { 2 } - 8 x + 11\).
Determine the value of the constant \(k\) for which the equation \(x ^ { 2 } - 8 x + 11 = k\) has two equal roots.
\(2 \xrightarrow { \text { The points } } O\) and \(A\) have position vectors \(\left( \begin{array} { l } 0 0 0 \end{array} \right)\) and \(\left( \begin{array} { l } 6 0 8 \end{array} \right)\) respectively. The point \(P\) is such that \(\overrightarrow { O P } = k \overrightarrow { O A }\), where \(k\) is a non-zero constant.
Find, in terms of \(k\), the length of \(O P\).
Point \(B\) has position vector \(\left( \begin{array} { l } 1 2 3 \end{array} \right)\) and angle \(O P B\) is a right angle.