Chapter 11 - Vectors and Vector-Valued Functions - 11.9 Curvature and Normal Vectors - 11.9 Exercises - Page 852: 13

$${\bf{T}}\left( t \right) = \frac{{\left\langle {1,2\cos t, - 2\sin t} \right\rangle }}{{\sqrt 5 }}{\text{ and }}\kappa \left( t \right) = \frac{1}{5}$$

$$\eqalign{ & {\bf{r}}\left( t \right) = \left\langle {2t,4\sin t,4\cos t} \right\rangle \cr & {\text{Calculate }}{\bf{v}}\left( t \right) \cr & {\bf{v}}\left( t \right) = {\bf{r}}'\left( t \right) \cr & {\bf{v}}\left( t \right) = \frac{d}{{dt}}\left\langle {2t,4\sin t,4\cos t} \right\rangle \cr & {\bf{v}}\left( t \right) = \left\langle {2,4\cos t, - 4\sin t} \right\rangle \cr & {\text{Find the unit tangent vector }}{\bf{T}} \cr & {\bf{T}}\left( t \right) = \frac{{{\bf{v}}\left( t \right)}}{{\left| {{\bf{v}}\left( t \right)} \right|}} \cr & {\bf{T}}\left( t \right) = \frac{{\left\langle {2,4\cos t, - 4\sin t} \right\rangle }}{{\left| {\left\langle {2, - 4\cos t, - 4\sin t} \right\rangle } \right|}} = \frac{{\left\langle {2,4\cos t, - 4\sin t} \right\rangle }}{{\sqrt {4 + 16{{\cos }^2}t + 16{{\sin }^2}t} }} \cr & {\bf{T}}\left( t \right) = \frac{{\left\langle {2,4\cos t, - 4\sin t} \right\rangle }}{{\sqrt {20} }} \cr & {\bf{T}}\left( t \right) = \frac{{\left\langle {2,4\cos t, - 4\sin t} \right\rangle }}{{2\sqrt 5 }} \cr & {\bf{T}}\left( t \right) = \frac{{\left\langle {1,2\cos t, - 2\sin t} \right\rangle }}{{\sqrt 5 }} \cr & \cr & {\text{Therefore}}{\text{, the curvature is}} \cr & \kappa \left( t \right) = \frac{1}{{\left| {\bf{v}} \right|}}\left| {\frac{{d{\bf{T}}}}{{dt}}} \right| \cr & \frac{{d{\bf{T}}}}{{dt}} = \frac{1}{{\sqrt 5 }}\frac{d}{{dt}}\left\langle {1,2\cos t, - 2\sin t} \right\rangle \cr & \frac{{d{\bf{T}}}}{{dt}} = \frac{1}{{\sqrt 5 }}\left\langle {0, - 2\sin t, - 2\cos t} \right\rangle \cr & \kappa \left( t \right) = \frac{1}{{\sqrt {20} }}\left| {\frac{1}{{\sqrt 5 }}\left\langle {0, - 2\sin t, - 2\cos t} \right\rangle } \right| \cr & \kappa \left( t \right) = \frac{1}{{\sqrt {100} }}\sqrt {0 + 4{{\sin }^2}t + 4{{\cos }^2}t} \cr & \kappa \left( t \right) = \frac{1}{{10}}\sqrt {4\left( {{{\sin }^2}t + {{\cos }^2}t} \right)} \cr & \kappa \left( t \right) = \frac{2}{{10}} \cr & \kappa \left( t \right) = \frac{1}{5} \cr & \cr & {\bf{T}}\left( t \right) = \frac{{\left\langle {1,2\cos t, - 2\sin t} \right\rangle }}{{\sqrt 5 }}{\text{ and }}\kappa \left( t \right) = \frac{1}{5} \cr} $$