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 \(\vec{v}(t)\) from the position function \(\vec{r}(t)\), you need to take the derivative of each component of the position vector with respect to time \(t\). Given the position function: \[ \vec{r}(t) = 4t\, \vec{\imath} + 4t\, \vec{\jmath} + 2t\, \vec{k} \] Compute the derivative of each component: - **For the \(\vec{\imath}\) component:** \[ \frac{d}{dt}(4t) = 4 \] - **For the \(\vec{\jmath}\) component:** \[ \frac{d}{dt}(4t) = 4 \] - **For the \(\vec{k}\) component:** \[ \frac{d}{dt}(2t) = 2 \] Putting it all together, the velocity function is: \[ \vec{v}(t) = 4\, \vec{\imath} + 4\, \vec{\jmath} + 2\, \vec{k} \] **Final Answer:** \[ \vec{v}(t) = 4\, \vec{\imath} + 4\, \vec{\jmath} + 2\, \vec{k} \]