Chapter 13: Vector-Valued Functions and Motion in Space - Section 13.4 - Curvature and Normal Vectors of a Curve - Exercises 13.4 - Page 765: 11

$${\bf{T}} = \left( {\frac{{\cos t - \sin t}}{{\sqrt 2 }}} \right){\bf{i}} + \left( {\frac{{\cos t + \sin t}}{{\sqrt 2 }}} \right){\bf{j}},{\bf{N}} = - \left( {\frac{{\sin t + \cos t}}{{\sqrt 2 }}} \right){\bf{i}} + \left( {\frac{{\cos t - \sin t}}{{\sqrt 2 }}} \right){\bf{j}},\,\kappa = \frac{1}{{\sqrt 2 {e^t}}}$$

$$\eqalign{ & {\bf{r}}\left( t \right) = \left( {{e^t}\cos t} \right){\bf{i}} + \left( {{e^t}\sin t} \right){\bf{j}} + 2{\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( {{e^t}\cos t} \right){\bf{i}} + \left( {{e^t}\sin t} \right){\bf{j}} + 2{\bf{k}}} \right] \cr & {\bf{v}}\left( t \right) = \left( { - {e^t}\sin t + {e^t}\cos t} \right){\bf{i}} + \left( {{e^t}\cos t + {e^t}\sin t} \right){\bf{j}} + 0{\bf{k}} \cr & {\bf{v}}\left( t \right) = \left( { - {e^t}\sin t + {e^t}\cos t} \right){\bf{i}} + \left( {{e^t}\cos t + {e^t}\sin t} \right){\bf{j}} \cr & \cr & {\text{We calculate }}{\bf{T}}{\text{ from the velocity vector}} \cr & {\bf{v}}\left( t \right) = \left( { - {e^t}\sin t + {e^t}\cos t} \right){\bf{i}} + \left( {{e^t}\cos t + {e^t}\sin t} \right){\bf{j}} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {{{\left( { - {e^t}\sin t + {e^t}\cos t} \right)}^2} + {{\left( {{e^t}\cos t + {e^t}\sin t} \right)}^2}} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {{e^{2t}}\left( {{{\cos }^2}t - 2\sin t\cos t + {{\sin }^2}t} \right) + {e^{2t}}\left( {{{\cos }^2}t + 2\sin t\cos t + {{\sin }^2}t} \right)} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {{e^{2t}}\left( {1 - 2\sin t\cos t} \right) + {e^{2t}}\left( {1 + 2\sin t\cos t} \right)} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {{e^{2t}} - 2{e^{2t}}\sin t\cos t + {e^{2t}} + 2{e^{2t}}\sin t\cos t} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {2{e^{2t}}} \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt 2 {e^t} \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{{\left( { - {e^t}\sin t + {e^t}\cos t} \right){\bf{i}} + \left( {{e^t}\cos t + {e^t}\sin t} \right){\bf{j}}}}{{\sqrt 2 {e^t}}} \cr & {\bf{T}} = \left( {\frac{{\cos t - \sin t}}{{\sqrt 2 }}} \right){\bf{i}} + \left( {\frac{{\cos t + \sin t}}{{\sqrt 2 }}} \right){\bf{j}} \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( {\left( {\frac{{\cos t - \sin t}}{{\sqrt 2 }}} \right){\bf{i}} + \left( {\frac{{\cos t + \sin t}}{{\sqrt 2 }}} \right){\bf{j}}} \right) \cr & \frac{{d{\bf{T}}}}{{dt}} = \left( {\frac{{ - \sin t - \cos t}}{{\sqrt 2 }}} \right){\bf{i}} + \left( {\frac{{ - \sin t + \cos t}}{{\sqrt 2 }}} \right){\bf{j}} \cr & \frac{{d{\bf{T}}}}{{dt}} = - \left( {\frac{{\sin t + \cos t}}{{\sqrt 2 }}} \right){\bf{i}} + \left( {\frac{{\cos t - \sin t}}{{\sqrt 2 }}} \right){\bf{j}} \cr & \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \sqrt {{{\left( {\frac{1}{{\sqrt 2 }}} \right)}^2}{{\left( {\sin t + \cos t} \right)}^2} + {{\left( {\frac{1}{{\sqrt 2 }}} \right)}^2}{{\left( {\cos t - \sin t} \right)}^2}} \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \sqrt {\frac{1}{2}\left( {{{\sin }^2}t + 2\sin t\cos t + {{\cos }^2}t} \right) + \frac{1}{2}\left( {{{\cos }^2}t - 2\sin t\cos t + {{\sin }^2}t} \right)} \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \sqrt {\frac{1}{2}\left( {1 + 2\sin t\cos t} \right) + \frac{1}{2}\left( {1 - 2\sin t\cos t} \right)} \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = \sqrt {\frac{1}{2} + \frac{1}{2}} \cr & \left| {\frac{{d{\bf{T}}}}{{dt}}} \right| = 1 \cr & {\bf{N}} = \frac{{d{\bf{T}}/dt}}{{\left| {d{\bf{T}}/dt} \right|}} = \frac{{ - \left( {\frac{{\sin t + \cos t}}{{\sqrt 2 }}} \right){\bf{i}} + \left( {\frac{{\cos t - \sin t}}{{\sqrt 2 }}} \right){\bf{j}}}}{1} \cr & {\bf{N}} = - \left( {\frac{{\sin t + \cos t}}{{\sqrt 2 }}} \right){\bf{i}} + \left( {\frac{{\cos t - \sin t}}{{\sqrt 2 }}} \right){\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}{{\sqrt 2 {e^t}}}\left( 1 \right) \cr & \kappa = \frac{1}{{\sqrt 2 {e^t}}} \cr} $$