Chapter 12 - Vector-Valued Functions - 12.3 Exercises - Page 838: 23

$$\eqalign{ & {\bf{v}}\left( t \right) = - \sin t{\bf{i}} + \cos t{\bf{j}} + {\bf{k}} \cr & {\bf{r}}\left( t \right) = \cos t{\bf{i}} + \sin t{\bf{j}} + t{\bf{k}} \cr & {\bf{r}}\left( 2 \right) = \left( {\cos 2} \right){\bf{i}} + \left( {\sin 2} \right){\bf{j}} + 2{\bf{k}} \cr} $$

$$\eqalign{ & {\bf{a}}\left( t \right) = - \cos t{\bf{i}} - \sin t{\bf{j}},{\text{ }}{\bf{v}}\left( 0 \right) = {\bf{j}} + {\bf{k}},{\text{ }}{\bf{r}}\left( 0 \right) = {\bf{i}} \cr & {\text{The velocity vector is}} \cr & {\bf{v}}\left( t \right) = \int {{\bf{a}}\left( t \right)} dt \cr & {\bf{v}}\left( t \right) = \int {\left( { - \cos t{\bf{i}} - \sin t{\bf{j}}} \right)} dt \cr & {\bf{v}}\left( t \right) = - \sin t{\bf{i}} + \cos t{\bf{j}} + {\bf{C}} \cr & {\text{Applying the initial condition }}{\bf{v}}\left( 0 \right) = {\bf{j}} + {\bf{k}} \cr & {\bf{v}}\left( 0 \right) = - \sin \left( 0 \right){\bf{i}} + \cos \left( 0 \right){\bf{j}} + {\bf{C}} \cr & {\bf{v}}\left( 0 \right) = 0{\bf{i}} + {\bf{j}} + {\bf{C}} \cr & {\bf{j}} + {\bf{k}} = {\bf{j}} + {\bf{C}} \cr & {\bf{C}} = {\bf{k}} \cr & {\text{Therefore,}} \cr & {\bf{v}}\left( t \right) = - \sin t{\bf{i}} + \cos t{\bf{j}} + {\bf{k}} \cr & \cr & {\text{The position vector is}} \cr & {\bf{r}}\left( t \right) = \int {{\bf{v}}\left( t \right)} dt \cr & {\bf{r}}\left( t \right) = \int {\left[ { - \sin t{\bf{i}} + \cos t{\bf{j}} + {\bf{k}}} \right]} dt \cr & {\bf{r}}\left( t \right) = \cos t{\bf{i}} + \sin t{\bf{j}} + t{\bf{k}} + {\bf{C}} \cr & {\text{Applying the initial condition }}{\bf{r}}\left( 0 \right) = {\bf{i}} \cr & {\bf{r}}\left( 0 \right) = \cos \left( 0 \right){\bf{i}} + \sin \left( 0 \right){\bf{j}} + \left( 0 \right){\bf{k}} + {\bf{C}} \cr & {\bf{r}}\left( 0 \right) = {\bf{i}} + 0{\bf{j}} + 0{\bf{k}} + {\bf{C}} \cr & {\bf{i}} = {\bf{i}} + {\bf{C}} \cr & {\bf{C}} = {\bf{0}} \cr & {\text{Therefore, the position vector is}} \cr & {\bf{r}}\left( t \right) = \cos t{\bf{i}} + \sin t{\bf{j}} + t{\bf{k}} \cr & \cr & {\text{Find the position at time }}t = 2 \cr & {\bf{r}}\left( 2 \right) = \cos \left( 2 \right){\bf{i}} + \sin \left( 2 \right){\bf{j}} + \left( 2 \right){\bf{k}} \cr & {\bf{r}}\left( 2 \right) = \left( {\cos 2} \right){\bf{i}} + \left( {\sin 2} \right){\bf{j}} + 2{\bf{k}} \cr} $$