Answer
\[\kappa = \frac{1}{3}\]
Work Step by Step
\[\begin{gathered}
{\mathbf{r}}\left( t \right) = \left\langle { - 3\cos t,3\sin t,0} \right\rangle \hfill \\
\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 { - 3\cos t,3\sin t,0} \right\rangle \hfill \\
{\mathbf{v}}\left( t \right) = \left\langle {3\sin t,3\cos t,0} \right\rangle \hfill \\
\left| {{\mathbf{v}}\left( t \right)} \right| = \sqrt {{{\left( {3\sin t} \right)}^2} + {{\left( {3\cos t} \right)}^2} + 0} \hfill \\
\left| {{\mathbf{v}}\left( t \right)} \right| = \sqrt {9{{\sin }^2}t + 9{{\cos }^2}t} \hfill \\
\left| {{\mathbf{v}}\left( t \right)} \right| = 3 \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 {3\sin t,3\cos t,0} \right\rangle \hfill \\
{\mathbf{a}}\left( t \right) = \left\langle {3\cos t, - 3\sin t,0} \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}} \\
{3\sin t}&{3\cos t}&0 \\
{3\cos t}&{ - 3\sin t}&0
\end{array}} \right| \hfill \\
{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right) = \left| {\begin{array}{*{20}{c}}
{3\cos t}&0 \\
{ - 3\sin t}&0
\end{array}} \right|{\mathbf{i}} - \left| {\begin{array}{*{20}{c}}
{3\sin t}&0 \\
{3\cos t}&0
\end{array}} \right|{\mathbf{j}} + \left| {\begin{array}{*{20}{c}}
{3\sin t}&{3\cos t} \\
{3\cos t}&{ - 3\sin t}
\end{array}} \right|{\mathbf{k}} \hfill \\
{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right) = 0{\mathbf{i}} - 0{\mathbf{j}} + \left( { - 9{{\sin }^2}t - 9{{\cos }^2}t} \right){\mathbf{k}} \hfill \\
{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right) = - 9\left( {{{\sin }^2}t + {{\cos }^2}t} \right){\mathbf{k}} \hfill \\
{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right) = - 9{\mathbf{k}} \hfill \\
\left| {{\mathbf{v}}\left( t \right) \times {\mathbf{a}}\left( t \right)} \right| = 9 \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{9}{{{{\left( 3 \right)}^3}}} \hfill \\
\kappa = \frac{1}{3} \hfill \\
\end{gathered} \]
