2 Fig. 4 shows the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) in the directions of the cartesian axes \(\mathrm { O } x\) and \(\mathrm { O } y\), respectively. O is the origin of the axes and of position vectors.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{496a9dfb-d330-4777-b5f9-a9d1b653dd7f-1_374_372_1431_911}
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\caption{Fig. 4}
\end{figure}
The position vector of a particle is given by \(\mathbf { r } = 3 t \mathbf { i } + \left( 18 t ^ { 2 } - 1 \right) \mathbf { j }\) for \(t \geqslant 0\), where \(t\) is time.
- Show that the path of the particle cuts the \(x\)-axis just once.
- Find an expression for the velocity of the particle at time \(t\).
Deduce that the particle never travels in the \(\mathbf { j }\) direction.
- Find the cartesian equation of the path of the particle, simplifying your answer.