5. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors and position vectors are given relative to a fixed origin \(O\).] A particle \(P\) is moving in a straight line with constant velocity. At 9 am, the position vector of \(P\) is \(( 7 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }\) and at 9.20 am , the position vector of \(P\) is \(6 \mathbf { i } \mathrm {~km}\). At time \(t\) hours after 9 am , the position vector of \(P\) is \(\mathbf { r } _ { P } \mathrm {~km}\).

1. Find, in \(\mathrm { kmh } ^ { - 1 }\), the speed of \(P\).
2. Show that \(\mathbf { r } _ { P } = ( 7 - 3 t ) \mathbf { i } + ( 5 - 15 t ) \mathbf { j }\).
3. Find the value of \(t\) when \(\mathbf { r } _ { P }\) is parallel to \(16 \mathbf { i } + 5 \mathbf { j }\). The position vector of another particle \(Q\), at time \(t\) hours after 9 am , is \(\mathbf { r } _ { Q } \mathrm {~km}\), where \(\mathbf { r } _ { Q } = ( 5 + 2 t ) \mathbf { i } + ( - 3 + 5 t ) \mathbf { j }\)
4. Show that \(P\) and \(Q\) will collide and find the position vector of the point of collision.