- (i) Relative to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation
$$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r }
- 5
1
6
\end{array} \right) + \lambda \left( \begin{array} { r }
2
- 3
1
\end{array} \right) \text { where } \lambda \text { is a scalar parameter. }$$
The point \(P\) lies on \(l _ { 1 }\). Given that \(\overrightarrow { O P }\) is perpendicular to \(l _ { 1 }\), calculate the coordinates of \(P\).
(ii) Relative to a fixed origin \(O\), the line \(l _ { 2 }\) is given by the equation
$$l _ { 2 } : \mathbf { r } = \left( \begin{array} { r }
4
- 3
12
\end{array} \right) + \mu \left( \begin{array} { r }
5
- 3
4
\end{array} \right) \text { where } \mu \text { is a scalar parameter. }$$
The point \(A\) does not lie on \(l _ { 2 }\). Given that the vector \(\overrightarrow { O A }\) is parallel to the line \(l _ { 2 }\) and \(| \overrightarrow { O A } | = \sqrt { 2 }\) units, calculate the possible position vectors of the point \(A\).