The position vector along a trajectory expressed in terms of the scalar distance \(s\) from the origin is given by the relation \(\mathbf{r}=\mathbf{a} s^{2}+\mathbf{b} s+\mathbf{c}\), with the condition that \(\mathbf{a}=0\) Determine the dimensions of the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) and show that the trajectory is that of a branch of a parabola passing through \(\mathbf{c}\). Draw the trajectory.