Chapter 12 - Vector-Valued Functions - Review Exercises - Page 864: 46

$$\eqalign{ & {\text{Parametric equations: }}x = 2 + t,{\text{ }}y = 4 + 4t,{\text{ }}z = \frac{{16}}{3} + 8t \cr & a = 1,{\text{ }}b = 4,{\text{ }}c = 8 \cr} $$

$$\eqalign{ & {\bf{r}}\left( t \right) = t{\bf{i}} + {t^2}{\bf{j}} + \frac{2}{3}{t^3}{\bf{k}},{\text{ }}P\left( {2,4,\frac{{16}}{3}} \right) \cr & {\text{Let }}t = 2 \cr & {\bf{r}}\left( 2 \right) = 2{\bf{i}} + {\left( 2 \right)^2}{\bf{j}} + \frac{2}{3}{\left( 2 \right)^3}{\bf{k}} \cr & {\bf{r}}\left( 2 \right) = 2{\bf{i}} + 4{\bf{j}} + \frac{{16}}{3}{\bf{k}},{\text{ then at }}P\left( {2,4,\frac{{16}}{3}} \right) \to {\text{ }}t = 2 \cr & {\text{Calculate }}{\bf{r}}{\text{'}}\left( t \right) \cr & {\bf{r}}'\left( t \right) = \frac{d}{{dt}}\left[ {t{\bf{i}} + {t^2}{\bf{j}} + \frac{2}{3}{t^3}{\bf{k}}} \right] \cr & {\bf{r}}'\left( t \right) = {\bf{i}} + 2t{\bf{j}} + 2{t^2}{\bf{k}} \cr & {\text{At }}t = 2 \cr & {\bf{r}}'\left( 2 \right) = {\bf{i}} + 2\left( 2 \right){\bf{j}} + 2{\left( 2 \right)^2}{\bf{k}} \cr & {\bf{r}}'\left( 2 \right) = {\bf{i}} + 4{\bf{j}} + 8{\bf{k}} \cr & {\text{at }}t = 2 \cr & {\bf{T}}\left( 2 \right) = \frac{{{\bf{r}}'\left( 2 \right)}}{{\left\| {{\bf{r}}{\text{'}}\left( 2 \right)} \right\|}} = \frac{{{\bf{i}} + 4{\bf{j}} + 8{\bf{k}}}}{{\left\| {{\bf{i}} + 4{\bf{j}} + 8{\bf{k}}} \right\|}} = \frac{{{\bf{i}} + 4{\bf{j}} + 8{\bf{k}}}}{{1 + 16 + 64}} \cr & {\bf{T}}\left( 2 \right) = \frac{1}{{81}}\left( {{\bf{i}} + 4{\bf{j}} + 8{\bf{k}}} \right) \cr & {\text{The direction numbers are}} \cr & a = 1,{\text{ }}b = 4,{\text{ }}c = 8 \cr & {\text{We can obtain the parametric equations:}} \cr & x = {x_1} + at \cr & y = {y_1} + bt \cr & z = {z_1} + ct \cr & \left( {2,4,\frac{{16}}{3}} \right) = \left( {{x_1},{y_1},{z_1}} \right) \cr & x = 2 + t \cr & y = 4 + 4t \cr & z = \frac{{16}}{3} + 8t \cr & x = 2 + t,{\text{ }}y = 4 + 4t,{\text{ }}z = \frac{{16}}{3} + 8t \cr & \cr & {\text{Parametric equations: }}x = 2 + t,{\text{ }}y = 4 + 4t,{\text{ }}z = \frac{{16}}{3} + 8t \cr & a = 1,{\text{ }}b = 4,{\text{ }}c = 8 \cr} $$