4 Two boats, \(A\) and \(B\), are moving on straight courses with constant speeds. At noon, \(A\) and \(B\) have position vectors \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { km }\) and \(( - \mathbf { i } + \mathbf { j } ) \mathrm { km }\) respectively relative to a lighthouse. Thirty minutes later, the position vectors of \(A\) and \(B\) are ( \(- \mathbf { i } + 3 \mathbf { j }\) ) km and \(( 2 \mathbf { i } - \mathbf { j } ) \mathrm { km }\) respectively relative to the lighthouse.

1. Find the velocity of \(A\) relative to \(B\) in the form \(( m \mathbf { i } + n \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\), where \(m\) and \(n\) are integers.
2. The position vector of \(A\) relative to \(B\) at time \(t\) hours after noon is \(\mathbf { r } \mathrm { km }\). Show that $$\mathbf { r } = ( 2 - 10 t ) \mathbf { i } + ( 1 + 6 t ) \mathbf { j }$$
3. Determine the value of \(t\) when \(A\) and \(B\) are closest together.
4. Find the shortest distance between \(A\) and \(B\).