Given the position function, find the velocity function. \[ \vec{r}(t)=4 t \vec{\imath}+4 t \vec{\jmath}+2 t \vec{k} \] \( \vec{v}(t)=[?] \vec{\imath}+[\quad] \vec{\jmath}+[\quad] \vec{k} \)

To find the velocity function \(\vec{v}(t)\), we differentiate the position function \(\vec{r}(t)\) with respect to time \(t\). The position function is given by: \[ \vec{r}(t) = 4t \vec{\imath} + 4t \vec{\jmath} + 2t \vec{k} \] Calculating the derivative, we get: \[ \vec{v}(t) = \frac{d\vec{r}}{dt} = \frac{d}{dt}(4t) \vec{\imath} + \frac{d}{dt}(4t) \vec{\jmath} + \frac{d}{dt}(2t) \vec{k} \] This results in: \[ \vec{v}(t) = 4 \vec{\imath} + 4 \vec{\jmath} + 2 \vec{k} \] Therefore, the velocity function is: \(\vec{v}(t) = 4 \vec{\imath} + 4 \vec{\jmath} + 2 \vec{k}\)