Chapter 12 - Vector-Valued Functions - Review Exercises - Page 864: 45

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

$$\eqalign{ & {\bf{r}}\left( t \right) = 2\cos t{\bf{i}} + 2\sin t{\bf{j}} + t{\bf{k}},{\text{ }}P\left( {1,\sqrt 3 ,\frac{\pi }{3}} \right) \cr & {\text{Let }}t = \frac{\pi }{3} \cr & {\bf{r}}\left( {\frac{\pi }{3}} \right) = 2\cos \left( {\frac{\pi }{3}} \right){\bf{i}} + 2\sin \left( {\frac{\pi }{3}} \right){\bf{j}} + \left( {\frac{\pi }{3}} \right){\bf{k}} \cr & {\bf{r}}\left( {\frac{\pi }{3}} \right) = {\bf{i}} + \sqrt 3 {\bf{j}} + \frac{\pi }{3}{\bf{k}},{\text{ then at }}P\left( {1,\sqrt 3 ,\frac{\pi }{3}} \right) \to {\text{ }}t = \frac{\pi }{3} \cr & {\text{Calculate }}{\bf{r}}{\text{'}}\left( t \right) \cr & {\bf{r}}'\left( t \right) = \frac{d}{{dt}}\left[ {2\cos t{\bf{i}} + 2\sin t{\bf{j}} + t{\bf{k}}} \right] \cr & {\bf{r}}'\left( t \right) = - 2\sin t{\bf{i}} + 2\cos t{\bf{j}} + {\bf{k}} \cr & {\text{At }}t = \frac{\pi }{3} \cr & {\bf{r}}'\left( {\frac{\pi }{3}} \right) = - 2\sin \left( {\frac{\pi }{3}} \right){\bf{i}} + 2\cos \left( {\frac{\pi }{3}} \right){\bf{j}} + {\bf{k}} \cr & {\bf{r}}'\left( {\frac{\pi }{3}} \right) = - \sqrt 3 {\bf{i}} + {\bf{j}} + {\bf{k}} \cr & {\text{at }}t = \frac{\pi }{3} \cr & {\bf{T}}\left( {\frac{\pi }{3}} \right) = \frac{{{\bf{r}}'\left( {\frac{\pi }{3}} \right)}}{{\left\| {{\bf{r}}{\text{'}}\left( {\frac{\pi }{3}} \right)} \right\|}} = \frac{{ - \sqrt 3 {\bf{i}} + {\bf{j}} + {\bf{k}}}}{{\left\| { - \sqrt 3 {\bf{i}} + {\bf{j}} + {\bf{k}}} \right\|}} = \frac{{ - \sqrt 3 {\bf{i}} + {\bf{j}} + {\bf{k}}}}{5} \cr & {\bf{T}}\left( {\frac{\pi }{3}} \right) = \frac{1}{5}\left( { - \sqrt 3 {\bf{i}} + {\bf{j}} + {\bf{k}}} \right) \cr & {\text{The direction numbers are}} \cr & a = - \sqrt 3 ,{\text{ }}b = 1,{\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( {1,\sqrt 3 ,\frac{\pi }{3}} \right) = \left( {{x_1},{y_1},{z_1}} \right) \cr & x = 1 - \sqrt 3 t \cr & y = \sqrt 3 + t \cr & z = \frac{\pi }{3} + t \cr & \cr & {\text{Parametric equations: }}x = 1 - \sqrt 3 t,{\text{ }}y = \sqrt 3 + t,{\text{ }}z = \frac{\pi }{3} + t \cr & {\text{Direction numbers: }}a = - \sqrt 3 ,{\text{ }}b = 1,{\text{ }}c = 1 \cr} $$