Chapter 13: Vector-Valued Functions and Motion in Space - Section 13.4 - Curvature and Normal Vectors of a Curve - Exercises 13.4 - Page 765: 12

$${\bf{T}} = \frac{{12}}{{13}}\cos 2t{\bf{i}} - \frac{{12}}{{13}}\sin 2t{\bf{j}} + \frac{5}{{13}}{\bf{k}},\,\,\,\,\,\,{\bf{N}} = - \sin 2t{\bf{i}} - \cos 2t{\bf{j}},\,\,\,\,\,\,\,\kappa = \frac{{24}}{{169}}$$

$$\eqalign{ & {\bf{r}}\left( t \right) = \left( {6\sin 2t} \right){\bf{i}} + \left( {6\cos 2t} \right){\bf{j}} + 5t{\bf{k}} \cr & {\text{Calculate }}{\bf{v}}\left( t \right).{\text{ Use }}{\bf{v}}\left( t \right) = {\bf{r}}'\left( t \right) \cr & {\bf{v}}\left( t \right) = {\bf{r}}'\left( t \right) = \frac{d}{{dt}}\left[ {\left( {6\sin 2t} \right){\bf{i}} + \left( {6\cos 2t} \right){\bf{j}} + 5t{\bf{k}}} \right] \cr & {\bf{v}}\left( t \right) = 12\cos 2t{\bf{i}} - 12\sin 2t{\bf{j}} + 5{\bf{k}} \cr & \cr & {\text{We calculate }}{\bf{T}}{\text{ from the velocity vector}} \cr & {\bf{v}}\left( t \right) = 12\cos 2t{\bf{i}} - 12\sin 2t{\bf{j}} + 5{\bf{k}} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {{{\left( {12\cos 2t} \right)}^2} + {{\left( { - 12\sin 2t} \right)}^2} + {{\left( 5 \right)}^2}} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {144{{\cos }^2}2t + 144{{\sin }^2}2t + 25} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {144 + 25} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {169} \cr & \left| {{\bf{v}}\left( t \right)} \right| = 13 \cr & \cr & {\text{use }}{\bf{T}}\left( t \right) = \frac{{{\bf{v}}\left( t \right)}}{{\left| {{\bf{v}}\left( t \right)} \right|}} \cr & {\bf{T}} = \frac{{12\cos 2t{\bf{i}} - 12\sin 2t{\bf{j}} + 5{\bf{k}}}}{{13}} \cr & {\bf{T}} = \frac{{12}}{{13}}\cos 2t{\bf{i}} - \frac{{12}}{{13}}\sin 2t{\bf{j}} + \frac{5}{{13}}{\bf{k}} \cr & \cr & {\text{Calculate }}{\bf{N}}\left( t \right){\text{ using the equation }}{\bf{N}} = \frac{{d{\bf{T}}/dt}}{{\left| {d{\bf{T}}/dt} \right|}}{\text{ }}\left( {{\text{see page 764}}} \right).{\text{ Then}} \cr & \frac{{d{\bf{T}}}}{{dt}} = \frac{d}{{dt}}\left( {\frac{{12}}{{13}}\cos 2t{\bf{i}} - \frac{{12}}{{13}}\sin 2t{\bf{j}} + \frac{5}{{13}}{\bf{k}}} \right) \cr & \frac{{d{\bf{T}}}}{{dt}} = - \frac{{24}}{{13}}\sin 2t{\bf{i}} - \frac{{24}}{{13}}\cos 2t{\bf{j}} \cr & \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \sqrt {{{\left( { - \frac{{24}}{{13}}\sin 2t} \right)}^2} + {{\left( { - \frac{{24}}{{13}}\cos 2t} \right)}^2}} \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \sqrt {{{\left( { - \frac{{24}}{{13}}} \right)}^2}\left( {{{\sin }^2}2t + {{\cos }^2}2t} \right)} \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \frac{{24}}{{13}} \cr & \cr & {\bf{N}} = \frac{{d{\bf{T}}/dt}}{{\left| {d{\bf{T}}/dt} \right|}} = \frac{{ - \frac{{24}}{{13}}\sin 2t{\bf{i}} - \frac{{24}}{{13}}\cos 2t{\bf{j}}}}{{24/13}} \cr & {\bf{N}} = - \sin 2t{\bf{i}} - \cos 2t{\bf{j}} \cr & \cr & {\text{Calculate }}\kappa {\text{ using the equation }}\kappa = \frac{1}{{\left| {\bf{v}} \right|}}\left| {\frac{{d{\bf{T}}}}{{dt}}} \right|{\text{ }}\left( {{\text{see page 764}}} \right).{\text{ Then}} \cr & \kappa = \frac{1}{{\left| {\bf{v}} \right|}}\left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \frac{1}{{13}}\left( {\frac{{24}}{{13}}} \right) \cr & \kappa = \frac{{24}}{{169}} \cr} $$