6. Two girls, Agatha and Brionie, are roller skating inside a large empty building. The girls are modelled as particles. At time \(t = 0\), Agatha is at the point with position vector \(( 11 \mathbf { i } + 11 \mathbf { j } ) \mathrm { m }\) and Brionie is at the point with position vector \(( 7 \mathbf { i } + 16 \mathbf { j } ) \mathrm { m }\). The position vectors are given relative to the door, \(O\), and \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors. Agatha skates with constant velocity ( \(3 \mathbf { i } - \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\)
Brionie skates with constant velocity ( \(4 \mathbf { i } - 2 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\)

1. Find the position vector of Agatha at time \(t\) seconds. At time \(t = 6\) seconds, Agatha passes through the point \(P\).
2. Show that Brionie also passes through \(P\) and find the value of \(t\) when this occurs. At time \(t\) seconds, Agatha is at the point \(A\) and Brionie is at the point \(B\).
3. Show that \(\overrightarrow { A B } = [ ( t - 4 ) \mathbf { i } + ( 5 - t ) \mathbf { j } ] \mathrm { m }\)
4. Find the distance between the two girls when they are closest together. \includegraphics[max width=\textwidth, alt={}, center]{ca445c1e-078c-4a57-94df-de90f30f8efd-13_2255_50_314_34}
   

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