Answer
\[{\mathbf{B}} = \frac{{\left\langle {5,12\sin t, - 12\cos t} \right\rangle }}{{13}}\]
\[\tau = \frac{{12}}{{169}}\]
Work Step by Step
\[\begin{gathered}
{\mathbf{r}}\left( t \right) = \left\langle {12t,5\cos t,5\sin t} \right\rangle \hfill \\
{\text{Calculate }}{\mathbf{v}}\left( t \right){\text{, }}{\mathbf{a}}\left( t \right){\text{ and }}{\mathbf{a}}'\left( t \right) \hfill \\
{\mathbf{v}}\left( t \right) = {\mathbf{r}}'\left( t \right) \hfill \\
{\mathbf{v}}\left( t \right) = \left\langle {12, - 5\sin t,5\cos t} \right\rangle \hfill \\
{\mathbf{a}}\left( t \right) = {\mathbf{v}}'\left( t \right) \hfill \\
{\mathbf{a}}\left( t \right) = \left\langle {0, - 5\cos t, - 5\sin t} \right\rangle \hfill \\
{\mathbf{a}}'\left( t \right) = \left\langle {0,5\sin t, - 5\cos t} \right\rangle \hfill \\
\hfill \\
{\text{Calculate }}{\mathbf{v}} \times {\mathbf{a}}{\text{ and }}\left| {{\mathbf{v}} \times {\mathbf{a}}} \right| \hfill \\
{\mathbf{v}} \times {\mathbf{a}} = \left| {\begin{array}{*{20}{c}}
{\mathbf{i}}&{\mathbf{j}}&{\mathbf{k}} \\
{12}&{ - 5\sin t}&{5\cos t} \\
0&{ - 5\cos t}&{ - 5\sin t}
\end{array}} \right| \hfill \\
{\mathbf{v}} \times {\mathbf{a}} = \left( {25{{\sin }^2}t + 25{{\cos }^2}t} \right){\mathbf{i}} + 60\sin t{\mathbf{j}} - 60\cos t{\mathbf{k}} \hfill \\
{\mathbf{v}} \times {\mathbf{a}} = 25{\mathbf{i}} + 60\sin t{\mathbf{j}} - 60\cos t{\mathbf{k}} \hfill \\
\left| {{\mathbf{v}} \times {\mathbf{a}}} \right| = \sqrt {{{25}^2} + {{60}^2}{{\sin }^2}t + {{60}^2}{{\cos }^2}t} \hfill \\
\left| {{\mathbf{v}} \times {\mathbf{a}}} \right| = \sqrt {4225} = 65 \hfill \\
\hfill \\
{\text{Calculate the unit binormal vector:}}\,{\text{ }}{\mathbf{B}} = \frac{{{\mathbf{v}} \times {\mathbf{a}}}}{{\left| {{\mathbf{v}} \times {\mathbf{a}}} \right|}} \hfill \\
{\mathbf{B}} = \frac{{{\mathbf{v}} \times {\mathbf{a}}}}{{\left| {{\mathbf{v}} \times {\mathbf{a}}} \right|}} = \frac{{\left\langle {25,60\sin t, - 60\cos t} \right\rangle }}{{65}} \hfill \\
{\mathbf{B}} = \frac{{\left\langle {5,12\sin t, - 12\cos t} \right\rangle }}{{13}} \hfill \\
\hfill \\
{\text{Calculate }}\tau \hfill \\
\tau = \frac{{\left( {{\mathbf{v}} \times {\mathbf{a}}} \right) \cdot {\mathbf{a}}'}}{{{{\left| {{\mathbf{v}} \times {\mathbf{a}}} \right|}^2}}} \hfill \\
\tau = \frac{{\left\langle {25,60\sin t, - 60\cos t} \right\rangle \cdot \left\langle {0,5\sin t, - 5\cos t} \right\rangle }}{{{{\left( {65} \right)}^2}}} \hfill \\
\tau = \frac{{0 + 300{{\sin }^2}t + 300{{\cos }^2}t}}{{4225}} = \frac{{300}}{{4225}} \hfill \\
\tau = \frac{{12}}{{169}} \hfill \\
\end{gathered} \]
