Chapter 12 - Vector-Valued Functions - 12.4 Exercises - Page 848: 9

$$\eqalign{ & {\text{Parametric equations: }}x = 3,{\text{ }}y = 3t,{\text{ }}z = t \cr & {\text{Direction numbers: }}a = 0,{\text{ }}b = 3,{\text{ }}c = 1 \cr} $$

$$\eqalign{ & {\bf{r}}\left( t \right) = 3\cos t{\bf{i}} + 3\sin t{\bf{j}} + t{\bf{k}},{\text{ }}P\left( {3,0,0} \right) \cr & {\text{Let }}t = 0 \cr & {\bf{r}}\left( 0 \right) = 3\cos \left( 0 \right){\bf{i}} + 3\sin \left( 0 \right){\bf{j}} + \left( 0 \right){\bf{k}} \cr & {\bf{r}}\left( 0 \right) = 3{\bf{i}} + 0{\bf{j}} + 0{\bf{k}},{\text{ then at }}P\left( {3,0,0} \right) \to {\text{ }}t = 0 \cr & {\text{Calculate }}{\bf{r}}{\text{'}}\left( t \right) \cr & {\bf{r}}'\left( t \right) = \frac{d}{{dt}}\left[ {3\cos t{\bf{i}} + 3\sin t{\bf{j}} + t{\bf{k}}} \right] \cr & {\bf{r}}'\left( t \right) = - 3\sin t{\bf{i}} + 3\cos t{\bf{j}} + {\bf{k}} \cr & {\text{At }}t = 0 \cr & {\bf{r}}'\left( 0 \right) = - 3\sin \left( 0 \right){\bf{i}} + 3\cos \left( 0 \right){\bf{j}} + {\bf{k}} \cr & {\bf{r}}'\left( 0 \right) = 0{\bf{i}} + 3{\bf{j}} + {\bf{k}} \cr & {\text{at }}t = 0 \cr & {\bf{T}}\left( 0 \right) = \frac{{{\bf{r}}'\left( 0 \right)}}{{\left\| {{\bf{r}}{\text{'}}\left( 0 \right)} \right\|}} = \frac{{0{\bf{i}} + 3{\bf{j}} + {\bf{k}}}}{{\left\| {0{\bf{i}} + 3{\bf{j}} + {\bf{k}}} \right\|}} = \frac{{0{\bf{i}} + 3{\bf{j}} + {\bf{k}}}}{{\sqrt {10} }} \cr & {\bf{T}}\left( 0 \right) = \frac{1}{{\sqrt {10} }}\left( {0{\bf{i}} + 3{\bf{j}} + {\bf{k}}} \right) \cr & {\text{The direction numbers are}} \cr & a = 0,{\text{ }}b = 3,{\text{ }}c = 1 \cr & {\text{We can obtain the parametric equations:}} \cr & x = {x_1} + at \cr & y = {y_1} + bt \cr & z = {z_1} + ct \cr & \left( {3,0,0} \right) = \left( {{x_1},{y_1},{z_1}} \right) \cr & x = 3 + 0t \cr & y = 0 + 3t \cr & z = 0 + t \cr & x = 3,{\text{ }}y = 3t,{\text{ }}z = t \cr & \cr & {\text{Parametric equations: }}x = 3,{\text{ }}y = 3t,{\text{ }}z = t \cr & {\text{Direction numbers: }}a = 0,{\text{ }}b = 3,{\text{ }}c = 1 \cr} $$