Chapter 12 - Vector-Valued Functions - 12.4 Exercises - Page 848: 11

$$\eqalign{ & {\text{Parametric equations: }}x = \sqrt 2 - \sqrt 2 t,{\text{ }}y = \sqrt 2 + \sqrt 2 t,{\text{ }}z = 0 \cr & {\text{Direction numbers: }}a = - \sqrt 2 ,{\text{ }}b = \sqrt 2 ,{\text{ }}c = 0 \cr} $$

$$\eqalign{ & {\bf{r}}\left( t \right) = \left\langle {2\cos t,2\sin t,4} \right\rangle ,{\text{ }}P\left( {\sqrt 2 ,\sqrt 2 ,4} \right) \cr & {\text{Let }}t = \frac{\pi }{4} \cr & {\bf{r}}\left( {\frac{\pi }{4}} \right) = \left\langle {2\cos \left( {\frac{\pi }{4}} \right),2\sin \left( {\frac{\pi }{4}} \right),4} \right\rangle \cr & {\bf{r}}\left( {\frac{\pi }{4}} \right) = \left\langle {\sqrt 2 ,\sqrt 2 ,4} \right\rangle ,{\text{ then at }}P\left( {\sqrt 2 ,\sqrt 2 ,4} \right) \to {\text{ }}t = \frac{\pi }{4} \cr & {\text{Calculate }}{\bf{r}}{\text{'}}\left( t \right) \cr & {\bf{r}}'\left( t \right) = \frac{d}{{dt}}\left[ {\left\langle {2\cos t,2\sin t,4} \right\rangle } \right] \cr & {\bf{r}}'\left( t \right) = \left\langle { - 2\sin t,2\cos t,0} \right\rangle \cr & {\text{At }}t = \frac{\pi }{4} \cr & {\bf{r}}'\left( {\frac{\pi }{4}} \right) = \left\langle { - 2\sin \left( {\frac{\pi }{4}} \right),2\cos \left( {\frac{\pi }{4}} \right),0} \right\rangle \cr & {\bf{r}}'\left( {\frac{\pi }{4}} \right) = \left\langle { - \sqrt 2 ,\sqrt 2 ,0} \right\rangle \cr & {\text{at }}t = \frac{\pi }{4} \cr & {\bf{T}}\left( {\frac{\pi }{4}} \right) = \frac{{{\bf{r}}'\left( {\pi /4} \right)}}{{\left\| {{\bf{r}}{\text{'}}\left( {\pi /4} \right)} \right\|}} = \frac{{\left\langle { - \sqrt 2 ,\sqrt 2 ,0} \right\rangle }}{{\left\| {\left\langle { - \sqrt 2 ,\sqrt 2 ,0} \right\rangle } \right\|}} = \frac{{\left\langle { - \sqrt 2 ,\sqrt 2 ,0} \right\rangle }}{2} \cr & {\bf{T}}\left( {\frac{\pi }{4}} \right) = \frac{1}{2}\left\langle { - \sqrt 2 ,\sqrt 2 ,0} \right\rangle \cr & {\text{The direction numbers are}} \cr & a = - \sqrt 2 ,{\text{ }}b = \sqrt 2 ,{\text{ }}c = 0 \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( {\sqrt 2 ,\sqrt 2 ,4} \right) = \left( {{x_1},{y_1},{z_1}} \right) \cr & x = \sqrt 2 - \sqrt 2 t \cr & y = \sqrt 2 + \sqrt 2 t \cr & z = 4 - 0t \cr & \cr & {\text{Parametric equations: }}x = \sqrt 2 - \sqrt 2 t,{\text{ }}y = \sqrt 2 + \sqrt 2 t,{\text{ }}z = 0 \cr & {\text{Direction numbers: }}a = - \sqrt 2 ,{\text{ }}b = \sqrt 2 ,{\text{ }}c = 0 \cr} $$