Answer
\[\kappa = \frac{{\sqrt 2 }}{{3e}}\]
Work Step by Step
\[\begin{gathered}
{\mathbf{r}}\left( t \right) = \left\langle {{e^t}\cos t,{e^t}\sin t,{e^t}} \right\rangle \hfill \\
{\text{Calculate }}{\mathbf{v}}\left( t \right){\text{ and }}\left| {{\mathbf{v}}\left( t \right)} \right| \hfill \\
{\mathbf{v}}\left( t \right) = {\mathbf{r}}'\left( t \right) \hfill \\
{\mathbf{v}}\left( t \right) = \frac{d}{{dt}}\left\langle {{e^t}\cos t,{e^t}\sin t,{e^t}} \right\rangle \hfill \\
{\mathbf{v}}\left( t \right) = \left\langle {{e^t}\cos t - {e^t}\sin t,{e^t}\sin t + {e^t}\cos t,{e^t}} \right\rangle \hfill \\
\left| {{\mathbf{v}}\left( t \right)} \right| = \sqrt {{{\left( {{e^t}\cos t - {e^t}\sin t} \right)}^2} + {{\left( {{e^t}\sin t + {e^t}\cos t} \right)}^2} + {{\left( {{e^t}} \right)}^2}} \hfill \\
\left| {{\mathbf{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) + {e^{2t}}} \hfill \\
\left| {{\mathbf{v}}\left( t \right)} \right| = \sqrt {{e^{2t}} + {e^{2t}} + {e^{2t}}} \hfill \\
\left| {{\mathbf{v}}\left( t \right)} \right| = \sqrt 3 {e^t} \hfill \\
\hfill \\
{\text{Calculate }}{\mathbf{a}}\left( t \right) \hfill \\
{\mathbf{a}}\left( t \right) = {\mathbf{v}}'\left( t \right) \hfill \\
{\mathbf{a}}\left( t \right) = \frac{d}{{dt}}\left\langle {{e^t}\cos t - {e^t}\sin t,{e^t}\sin t + {e^t}\cos t,{e^t}} \right\rangle \hfill \\
{\mathbf{a}}\left( t \right) = \left\langle { - 2{e^t}\sin t,2{e^t}\cos t,{e^t}} \right\rangle \hfill \\
\hfill \\
{\text{Calculate }}\left| {{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right)} \right| \hfill \\
{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right) = \left| {\begin{array}{*{20}{c}}
{\mathbf{i}}&{\mathbf{j}}&{\mathbf{k}} \\
{{e^t}\cos t - {e^t}\sin t}&{{e^t}\cos t + {e^t}\sin t}&{{e^t}} \\
{ - 2{e^t}\sin t}&{2{e^t}\cos t}&{{e^t}}
\end{array}} \right| \hfill \\
{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right) = \left| {\begin{array}{*{20}{c}}
{{e^t}\cos t + {e^t}\sin t}&{{e^t}} \\
{2{e^t}\cos t}&{{e^t}}
\end{array}} \right|{\mathbf{i}} - \left| {\begin{array}{*{20}{c}}
{{e^t}\cos t - {e^t}\sin t}&{{e^t}} \\
{ - 2{e^t}\sin t}&{{e^t}}
\end{array}} \right|{\mathbf{j}} \hfill \\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left| {\begin{array}{*{20}{c}}
{{e^t}\cos t - {e^t}\sin t}&{{e^t}\cos t + {e^t}\sin t} \\
{ - 2{e^t}\sin t}&{2{e^t}\cos t}
\end{array}} \right|{\mathbf{k}} \hfill \\
{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right) = {e^{2t}}\left( {\sin t - \cos t} \right){\mathbf{i}} + {e^{2t}}\left( {\cos t + \sin t} \right){\mathbf{j}} + 2{e^{2t}}{\mathbf{k}} \hfill \\
\left| {{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right)} \right| = \sqrt {{e^{4t}}\left( {1 - 2\sin t\cos t} \right) + {e^{4t}}\left( {1 + 2\sin t\cos t} \right) + 4{e^{4t}}} \hfill \\
\left| {{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right)} \right| = \sqrt {{e^{4t}} + {e^{4t}} + 4{e^{4t}}} \hfill \\
\left| {{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right)} \right| = \sqrt 6 {e^{2t}} \hfill \\
\hfill \\
{\text{Use the alternative curvature formula}} \hfill \\
\kappa = \frac{{\left| {{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right)} \right|}}{{{{\left| {{\mathbf{v}}\left( t \right)} \right|}^3}}} \hfill \\
\kappa = \frac{{\sqrt 6 {e^{2t}}}}{{{{\left( {\sqrt 3 {e^t}} \right)}^3}}} \hfill \\
\kappa = \frac{{\sqrt 6 {e^{2t}}}}{{3\sqrt 3 {e^{3t}}}} \hfill \\
\kappa = \frac{{\sqrt 2 }}{{3e}} \hfill \\
\end{gathered} \]
