\hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively.] A hiker \(H\) is walking with constant velocity \(( 1.2 \mathbf { i } - 0.9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
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A horizontal field \(O A B C\) is rectangular with \(O A\) due east and \(O C\) due north, as shown in Figure 3. At twelve noon hiker \(H\) is at the point \(Y\) with position vector \(100 \mathbf { j } \mathrm {~m}\), relative to the fixed origin \(O\).
Write down the position vector of \(H\) at time \(t\) seconds after noon.
At noon, another hiker \(K\) is at the point with position vector \(( 9 \mathbf { i } + 46 \mathbf { j } )\) m. Hiker \(K\) is moving with constant velocity \(( 0.75 \mathbf { i } + 1.8 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
Show that, at time \(t\) seconds after noon,
$$\overrightarrow { H K } = [ ( 9 - 0.45 t ) \mathbf { i } + ( 2.7 t - 54 ) \mathbf { j } ] \text { metres. }$$
Hence,
show that the two hikers meet and find the position vector of the point where they meet.