10 Relative to an origin \(O\), the position vectors of points \(A , B\) and \(C\) are given by
$$\overrightarrow { O A } = \left( \begin{array} { r }
2
1
- 2
\end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r }
5
- 1
k
\end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r }
2
6
- 3
\end{array} \right)$$
respectively, where \(k\) is a constant.
- Find the value of \(k\) in the case where angle \(A O B = 90 ^ { \circ }\).
- Find the possible values of \(k\) for which the lengths of \(A B\) and \(O C\) are equal.
The point \(D\) is such that \(\overrightarrow { O D }\) is in the same direction as \(\overrightarrow { O A }\) and has magnitude 9 units. The point \(E\) is such that \(\overrightarrow { O E }\) is in the same direction as \(\overrightarrow { O C }\) and has magnitude 14 units.
- Find the magnitude of \(\overrightarrow { D E }\) in the form \(\sqrt { } n\) where \(n\) is an integer.