5. At time \(t = 0\) particles \(P\) and \(Q\) start simultaneously from points which have position vectors \(( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }\) and \(( - \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { m }\) respectively, relative to a fixed origin \(O\). The velocities of \(P\) and \(Q\) are \(( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and \(( 2 \mathbf { i } + \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) respectively.

1. Show that \(P\) and \(Q\) collide and find the position vector of the point at which they collide. A third particle \(R\) moves in such a way that its velocity relative to \(P\) is parallel to the vector ( \(- 5 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }\) ) and its velocity relative to \(Q\) is parallel to the vector \(( - 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } )\). Given that all three particles collide simultaneously, find
2. 1. the velocity of \(R\),
   2. the position vector of \(R\) at time \(t = 0\).