6. [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.] A ship sets sail at 9 am from a port \(P\) and moves with constant velocity. The position vector of \(P\) is \(( 4 \mathbf { i } - 8 \mathbf { j } ) \mathrm { km }\). At 9.30 am the ship is at the point with position vector \(( \mathbf { i } - 4 \mathbf { j } ) \mathrm { km }\).

1. Find the speed of the ship in \(\mathrm { km } \mathrm { h } ^ { - 1 }\).
2. Show that the position vector \(\mathbf { r } \mathrm { km }\) of the ship, \(t\) hours after 9 am , is given by \(\mathbf { r } = ( 4 - 6 t ) \mathbf { i } + ( 8 t - 8 ) \mathbf { j }\). At 10 am , a passenger on the ship observes that a lighthouse \(L\) is due west of the ship. At 10.30 am , the passenger observes that \(L\) is now south-west of the ship.
3. Find the position vector of \(L\).