6. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin.] At 2 pm , the position vector of ship \(P\) is \(( 5 \mathbf { i } - 3 \mathbf { j } ) \mathrm { km }\) and the position vector of ship \(Q\) is \(( 7 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }\).

1. Find the distance between \(P\) and \(Q\) at 2 pm . Ship \(P\) is moving with constant velocity \(( 2 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and ship \(Q\) is moving with constant velocity \(( - 3 \mathbf { i } - 15 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).
2. Find the position vector of \(P\) at time \(t\) hours after 2 pm .
3. Find the position vector of \(Q\) at time \(t\) hours after 2 pm .
4. Show that \(Q\) will meet \(P\) and find the time at which they meet.
5. Find the position vector of the point at which they meet.