Answer
False
Work Step by Step
$$\eqalign{
& {\text{Let }}{\bf{r}}\left( t \right) = \left( {3t - 1} \right){\bf{i}} + \frac{1}{4}{t^3}{\bf{j}} + 4{\bf{k}}{\text{ and }}{\bf{u}}\left( t \right) = {t^2}{\bf{i}} - 8{\bf{j}} + {t^3}{\bf{k}} \cr
& {\text{Find }}{\bf{r}}\left( t \right) \cdot {\bf{u}}\left( t \right) \cr
& {\bf{r}}\left( t \right) \cdot {\bf{u}}\left( t \right) = \left[ {\left( {3t - 1} \right){\bf{i}} + \frac{1}{4}{t^3}{\bf{j}} + 4{\bf{k}}} \right] \cdot \left[ {{t^2}{\bf{i}} - 8{\bf{j}} + {t^3}{\bf{k}}} \right] \cr
& {\bf{r}}\left( t \right) \cdot {\bf{u}}\left( t \right) = \left( {3t - 1} \right){t^2} + \frac{1}{4}{t^3}\left( { - 8} \right) + 4{t^3} \cr
& {\bf{r}}\left( t \right) \cdot {\bf{u}}\left( t \right) = 3{t^3} - {t^2} - 2{t^3} + 4{t^3} \cr
& {\bf{r}}\left( t \right) \cdot {\bf{u}}\left( t \right) = 5{t^3} - {t^2} \cr
& {\text{The statement is false, the dot product is a scalar - valued function}} \cr} $$
