Chapter 12 - Vector-Valued Functions - 12.3 Exercises - Page 838: 19

$$\eqalign{ & {\bf{v}}\left( t \right) = t\left( {{\bf{i}} + {\bf{j}} + {\bf{k}}} \right) \cr & {\bf{r}}\left( t \right) = \frac{1}{2}{t^2}{\bf{i}} + \frac{1}{2}{t^2}{\bf{j}} + \frac{1}{2}{t^2}{\bf{k}} \cr & {\bf{r}}\left( 2 \right) = 2{\bf{i}} + 2{\bf{j}} + 2{\bf{k}} \cr} $$

$$\eqalign{ & {\bf{a}}\left( t \right) = {\bf{i}} + {\bf{j}} + {\bf{k}},{\text{ }}{\bf{v}}\left( 0 \right) = {\bf{0}},{\text{ }}{\bf{r}}\left( 0 \right) = {\bf{0}} \cr & {\text{The velocity vector is}} \cr & {\bf{v}}\left( t \right) = \int {{\bf{a}}\left( t \right)} dt \cr & {\bf{v}}\left( t \right) = \int {\left( {{\bf{i}} + {\bf{j}} + {\bf{k}}} \right)} dt \cr & {\bf{v}}\left( t \right) = t{\bf{i}} + t{\bf{j}} + t{\bf{k}} + {\bf{C}} \cr & {\text{Applying the initial condition }}{\bf{v}}\left( 0 \right) = {\bf{0}} \cr & {\bf{v}}\left( 0 \right) = \left( 0 \right){\bf{i}} + \left( 0 \right){\bf{j}} + \left( 0 \right){\bf{k}} + {\bf{C}} \cr & {\bf{v}}\left( 0 \right) = {\bf{C}} \cr & {\bf{0}} = {\bf{C}} \cr & {\text{Therefore,}} \cr & {\bf{v}}\left( t \right) = t{\bf{i}} + t{\bf{j}} + t{\bf{k}} \cr & {\bf{v}}\left( t \right) = t\left( {{\bf{i}} + {\bf{j}} + {\bf{k}}} \right) \cr & \cr & {\text{The position vector is}} \cr & {\bf{r}}\left( t \right) = \int {{\bf{v}}\left( t \right)} dt \cr & {\bf{r}}\left( t \right) = \int {\left( {t{\bf{i}} + t{\bf{j}} + t{\bf{k}}} \right)} dt \cr & {\bf{r}}\left( t \right) = \frac{1}{2}{t^2}{\bf{i}} + \frac{1}{2}{t^2}{\bf{j}} + \frac{1}{2}{t^2}{\bf{k}} + {\bf{C}} \cr & {\text{Applying the initial condition }}{\bf{r}}\left( 0 \right) = {\bf{0}} \cr & {\bf{r}}\left( 0 \right) = \frac{1}{2}{\left( 0 \right)^2}{\bf{i}} + \frac{1}{2}{\left( 0 \right)^2}{\bf{j}} + \frac{1}{2}{\left( 0 \right)^2}{\bf{k}} + {\bf{C}} \cr & {\bf{r}}\left( 0 \right) = {\bf{C}} \cr & {\bf{0}} = {\bf{C}} \cr & {\text{Therefore, the position vector is}} \cr & {\bf{r}}\left( t \right) = \frac{1}{2}{t^2}{\bf{i}} + \frac{1}{2}{t^2}{\bf{j}} + \frac{1}{2}{t^2}{\bf{k}} \cr & \cr & {\text{Find the position at time }}t = 2 \cr & {\bf{r}}\left( 2 \right) = \frac{1}{2}{\left( 2 \right)^2}{\bf{i}} + \frac{1}{2}{\left( 2 \right)^2}{\bf{j}} + \frac{1}{2}{\left( 2 \right)^2}{\bf{k}} \cr & {\bf{r}}\left( 2 \right) = 2{\bf{i}} + 2{\bf{j}} + 2{\bf{k}} \cr} $$