A particle moves in a plane such that its position vector \(\vec{r}(t)\) as a function of time \(t\) is given by

\[\vec{r}(t) = b t^2\,\hat{e}_1 + c t\,\hat{e}_2\]

where \(b\) and \(c\) are positive constants, and \(\hat{e}_1\) and \(\hat{e}_2\) are fixed unit vectors in the plane separated by a constant angle \(\theta\) (where \(0 < \theta < \pi\)). Which of the following expressions represents the tangential acceleration \(a_t(t) = \dfrac{dv}{dt}\) of the particle as a function of time \(t\)?