8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f2737a11-4a15-41e9-9f87-31a705a8948b-22_792_841_246_612}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{figure}
A square floor space \(A B C D\), with centre \(O\), is modelled as a flat horizontal surface measuring 50 m by 50 m , as shown in Figure 5 .
The horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the direction of \(\overrightarrow { A B }\) and \(\overrightarrow { A D }\) respectively.
All position vectors are given relative to \(O\).
A small robot \(R\) is programmed to travel across the floor at a constant velocity.
- At time \(t = 0 , R\) is at the point with position vector ( \(- 2 \mathbf { i } + \mathbf { j }\) ) m
- At time \(t = 11 \mathrm {~s} , R\) is at the point with position vector \(( 9 \mathbf { i } + 23 \mathbf { j } ) \mathrm { m }\)
- At time \(t\) seconds, the position vector of \(R\) is \(\mathbf { r }\) metres
- Find, in terms of \(t\), i and j, an expression for \(\mathbf { r }\)
A second robot \(S\) is at the point \(C\).
- At time \(t = 0 , S\) leaves \(C\) and moves with constant velocity \(( - \mathbf { i } - \mathbf { j } ) \mathrm { ms } ^ { - 1 }\)
- At time \(t\) seconds, the position vector of \(S\) is \(\mathbf { s }\) metres
- Write down, in terms of \(t\), i and \(\mathbf { j }\), an expression for \(\mathbf { s }\)
- Show that
$$\overrightarrow { S R } = [ ( 2 t - 27 ) \mathbf { i } + ( 3 t - 24 ) \mathbf { j } ] \mathbf { m }$$
Find the time when the distance between \(R\) and \(S\) is a minimum.