Chapter 13: Vector-Valued Functions and Motion in Space - Section 13.4 - Curvature and Normal Vectors of a Curve - Exercises 13.4 - Page 765: 9

$${\bf{T}} = \frac{3}{5}\cos t{\bf{i}} - \frac{3}{5}\sin t{\bf{j}} + \frac{4}{5}{\bf{k}},\,\,\,\,\,\,\,\,\,{\bf{N}} = - \sin t{\bf{i}} - \cos t{\bf{j}},\,\,\,\,\,\,\,\,\,\kappa = \frac{3}{{25}}$$

$$\eqalign{ & {\bf{r}}\left( t \right) = \left( {3\sin t} \right){\bf{i}} + \left( {3\cos t} \right){\bf{j}} + 4t{\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( {3\sin t} \right){\bf{i}} + \left( {3\cos t} \right){\bf{j}} + 4t{\bf{k}}} \right] \cr & {\bf{v}}\left( t \right) = 3\cos t{\bf{i}} - 3\sin t{\bf{j}} + 4{\bf{k}} \cr & \cr & {\text{We calculate }}{\bf{T}}{\text{ from the velocity vector}} \cr & {\bf{v}}\left( t \right) = 3\cos t{\bf{i}} - 3\sin t{\bf{j}} + 4{\bf{k}} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {{{\left( {3\cos t} \right)}^2} + {{\left( { - 3\sin t} \right)}^2} + {{\left( 4 \right)}^2}} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {9{{\cos }^2}t + 9{{\sin }^2}t + 16} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {9\left( {{{\cos }^2}t + {{\sin }^2}t} \right) + 16} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {25} \cr & \left| {{\bf{v}}\left( t \right)} \right| = 5 \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{{3\cos t{\bf{i}} - 3\sin t{\bf{j}} + 4{\bf{k}}}}{5} \cr & {\bf{T}} = \frac{3}{5}\cos t{\bf{i}} - \frac{3}{5}\sin t{\bf{j}} + \frac{4}{5}{\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{3}{5}\cos t{\bf{i}} - \frac{3}{5}\sin t{\bf{j}} + \frac{4}{5}{\bf{k}}} \right) \cr & \frac{{d{\bf{T}}}}{{dt}} = - \frac{3}{5}\sin t{\bf{i}} - \frac{3}{5}\cos t{\bf{j}} \cr & \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \sqrt {{{\left( { - \frac{3}{5}\sin t} \right)}^2} + {{\left( { - \frac{3}{5}\cos t} \right)}^2}} \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \sqrt {\frac{9}{{25}}{{\sin }^2}t + \frac{9}{{25}}{{\cos }^2}t} \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \sqrt {\frac{9}{{25}}} = \frac{3}{5} \cr & {\bf{N}} = \frac{{d{\bf{T}}/dt}}{{\left| {d{\bf{T}}/dt} \right|}} = \frac{{ - \frac{3}{5}\sin t{\bf{i}} - \frac{3}{5}\cos t{\bf{j}}}}{{3/5}} \cr & {\bf{N}} = - \sin t{\bf{i}} - \cos t{\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}{5}\left( {\frac{3}{5}} \right) \cr & \kappa = \frac{3}{{25}} \cr} $$