8. A particle \(P\) moves in the \(x - y\) plane and has position vector \(\mathbf { r }\) metres relative to a fixed origin \(O\) at time \(t \mathrm {~s}\). Given that \(\mathbf { r }\) satisfies the vector differential equation $$\frac { \mathrm { d } ^ { 2 } \mathbf { r } } { \mathrm {~d} t ^ { 2 } } + 9 \mathbf { r } = 8 \sin t \mathbf { i }$$ and that when \(t = 0 \mathrm {~s} , P\) is at \(O\) and moving with velocity \(( \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\),

1. find \(\mathbf { r }\) at time \(t\).
2. Hence find when \(P\) next returns to \(O\).