Chapter 11 - Vectors and Vector-Valued Functions - 11.9 Curvature and Normal Vectors - 11.9 Exercises - Page 852: 44

$${\bf{B}} = \left\langle {0,0,1} \right\rangle {\text{ and }}\tau = 0$$

$$\eqalign{ & {\bf{r}}\left( t \right) = \left\langle {\frac{{{t^2}}}{2},\frac{{{t^3}}}{3}} \right\rangle ,\,\,\,t > 0 \cr & {\text{Calculate }}{\bf{v}}\left( t \right){\text{, }}{\bf{a}}\left( t \right){\text{ and }}{\bf{a}}'\left( t \right) \cr & {\bf{v}}\left( t \right) = {\bf{r}}'\left( t \right) \cr & {\bf{v}}\left( t \right) = \left\langle {t,{t^2}} \right\rangle \cr & {\bf{a}}\left( t \right) = {\bf{v}}'\left( t \right) \cr & {\bf{a}}\left( t \right) = \left\langle {1,2t} \right\rangle \cr & {\bf{a}}'\left( t \right) = \left\langle {0,2} \right\rangle \cr & \cr & {\text{Calculate }}{\bf{v}} \times {\bf{a}}{\text{ and }}\left| {{\bf{v}} \times {\bf{a}}} \right| \cr} $$ \[{\mathbf{v}} \times {\mathbf{a}} = \left| {\begin{array}{*{20}{c}} {\mathbf{i}}&{\mathbf{j}}&{\mathbf{k}} \\ t&{{t^2}}&0 \\ 1&{2t}&0 \end{array}} \right|\] $$\eqalign{ & {\bf{v}} \times {\bf{a}} = 0{\bf{i}} - 0{\bf{j}} + \left( {2{t^2} - {t^2}} \right){\bf{k}} \cr & {\bf{v}} \times {\bf{a}} = {t^2}{\bf{k}} \cr & \left| {{\bf{v}} \times {\bf{a}}} \right| = \sqrt {{t^4}} \cr & \left| {{\bf{v}} \times {\bf{a}}} \right| = {t^2} \cr & \cr & {\text{Calculate the unit binormal vector:}}\,{\text{ }}{\bf{B}} = \frac{{{\bf{v}} \times {\bf{a}}}}{{\left| {{\bf{v}} \times {\bf{a}}} \right|}} \cr & {\bf{B}} = \frac{{{\bf{v}} \times {\bf{a}}}}{{\left| {{\bf{v}} \times {\bf{a}}} \right|}} = \frac{{{t^2}{\bf{k}}}}{{{t^2}}} \cr & {\bf{B}} = \left\langle {0,0,1} \right\rangle \cr & \cr & {\text{Calculate }}\tau \cr & \tau = \frac{{\left( {{\bf{v}} \times {\bf{a}}} \right) \cdot {\bf{a}}'}}{{{{\left| {{\bf{v}} \times {\bf{a}}} \right|}^2}}} \cr & \tau = \frac{{\left\langle {0,0,{t^2}} \right\rangle \cdot \left\langle {0,2,0} \right\rangle }}{{{{\left( {{t^2}} \right)}^2}}} \cr & \tau = 0 \cr} $$