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

$$\eqalign{ & {\text{Parametric equations: }}x = 1 + \sqrt 3 t,{\text{ }}y = \sqrt 3 - t,{\text{ }}z = 1 + 2\sqrt 3 t \cr & {\text{Direction numbers: }}a = \sqrt 3 ,{\text{ }}b = - 1,{\text{ }}c = 2\sqrt 3 \cr} $$

$$\eqalign{ & {\bf{r}}\left( {\frac{\pi }{6}} \right) = \left\langle {2\sin \left( {\frac{\pi }{6}} \right),2\cos \left( {\frac{\pi }{6}} \right),4{{\sin }^2}\left( {\frac{\pi }{6}} \right)} \right\rangle \cr & {\bf{r}}\left( {\frac{\pi }{6}} \right) = \left\langle {1,\sqrt 3 ,1} \right\rangle ,{\text{ then at }}P\left( {1,\sqrt 3 ,1} \right) \to {\text{ }}t = \frac{\pi }{6} \cr & {\text{Calculate }}{\bf{r}}{\text{'}}\left( t \right) \cr & {\bf{r}}'\left( t \right) = \frac{d}{{dt}}\left[ {\left\langle {2\sin t,2\cos t,4{{\sin }^2}t} \right\rangle } \right] \cr & {\bf{r}}'\left( t \right) = \left\langle {2\cos t, - 2\sin t,8\sin t\cos t} \right\rangle \cr & {\bf{r}}'\left( t \right) = \left\langle {2\cos t, - 2\sin t,4\sin 2t} \right\rangle \cr & {\text{At }}t = \frac{\pi }{6} \cr & {\bf{r}}'\left( {\frac{\pi }{6}} \right) = \left\langle {2\cos \left( {\frac{\pi }{6}} \right), - 2\sin \left( {\frac{\pi }{6}} \right),4\sin 2\left( {\frac{\pi }{6}} \right)} \right\rangle \cr & {\bf{r}}'\left( {\frac{\pi }{6}} \right) = \left\langle {\sqrt 3 , - 1,2\sqrt 3 } \right\rangle \cr & {\text{at }}t = \frac{\pi }{6} \cr & {\bf{T}}\left( {\frac{\pi }{6}} \right) = \frac{{{\bf{r}}'\left( {\pi /6} \right)}}{{\left\| {{\bf{r}}{\text{'}}\left( {\pi /6} \right)} \right\|}} = \frac{{\left\langle {\sqrt 3 , - 1,2\sqrt 3 } \right\rangle }}{{\left\| {\left\langle {\sqrt 3 , - 1,2\sqrt 3 } \right\rangle } \right\|}} = \frac{{\left\langle {\sqrt 3 , - 1,2\sqrt 3 } \right\rangle }}{4} \cr & {\bf{T}}\left( {\frac{\pi }{6}} \right) = \frac{1}{4}\left\langle {\sqrt 3 , - 1,2\sqrt 3 } \right\rangle \cr & {\text{The direction numbers are}} \cr & a = \sqrt 3 ,{\text{ }}b = - 1,{\text{ }}c = 2\sqrt 3 \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 ,1} \right) = \left( {{x_1},{y_1},{z_1}} \right) \cr & x = 1 + \sqrt 3 t \cr & y = \sqrt 3 - t \cr & z = 1 + 2\sqrt 3 t \cr & \cr & {\text{Parametric equations: }}x = 1 + \sqrt 3 t,{\text{ }}y = \sqrt 3 - t,{\text{ }}z = 1 + 2\sqrt 3 t \cr & {\text{Direction numbers: }}a = \sqrt 3 ,{\text{ }}b = - 1,{\text{ }}c = 2\sqrt 3 \cr} $$