Answer
$$21\sin t\cos t$$
Work Step by Step
$$\eqalign{
& {\bf{r}}\left( t \right) = 5\cos t{\bf{i}} + 2\sin t{\bf{j}} \cr
& \left( {\text{a}} \right){\text{Calculate }}{\bf{r}}{\text{'}}\left( t \right) \cr
& {\bf{r}}{\text{'}}\left( t \right) = \frac{d}{{dt}}\left[ {5\cos t} \right]{\bf{i}} + \frac{d}{{dt}}\left[ {2\sin t} \right]{\bf{j}} \cr
& {\bf{r}}{\text{'}}\left( t \right) = - 5\sin t{\bf{i}} + 2\cos t{\bf{j}} \cr
& \cr
& \left( {\text{b}} \right){\text{Calculate }}{\bf{r}}{\text{''}}\left( t \right) \cr
& {\bf{r}}{\text{''}}\left( t \right) = \frac{d}{{dt}}\left[ {{\bf{r}}{\text{'}}\left( t \right)} \right] \cr
& {\bf{r}}{\text{''}}\left( t \right) = \frac{d}{{dt}}\left[ { - 5\sin t} \right]{\bf{i}} + \frac{d}{{dt}}\left[ {2\cos t} \right]{\bf{j}} \cr
& {\bf{r}}{\text{''}}\left( t \right) = - 5\cos t{\bf{i}} - 2\sin t{\bf{j}} \cr
& \cr
& \left( {\text{c}} \right){\text{Calculate the dot product }}{\bf{r}}{\text{'}}\left( t \right) \cdot {\bf{r}}{\text{''}}\left( t \right) \cr
& {\bf{r}}{\text{'}}\left( t \right) \cdot {\bf{r}}{\text{''}}\left( t \right) = \left[ { - 5\sin t{\bf{i}} + 2\cos t{\bf{j}}} \right] \cdot \left( { - 5\cos t{\bf{i}} - 2\sin t{\bf{j}}} \right) \cr
& {\bf{r}}{\text{'}}\left( t \right) \cdot {\bf{r}}{\text{''}}\left( t \right) = \left( { - 5\sin t} \right)\left( { - 5\cos t} \right) + \left( {2\cos t} \right)\left( { - 2\sin t} \right) \cr
& {\bf{r}}{\text{'}}\left( t \right) \cdot {\bf{r}}{\text{''}}\left( t \right) = 25\sin t\cos t - 4\sin t\cos t \cr
& {\bf{r}}{\text{'}}\left( t \right) \cdot {\bf{r}}{\text{''}}\left( t \right) = 21\sin t\cos t \cr} $$
