The torsion of a curve defined by $\mathbf { r } ( t )$ is given by
$$\tau = \frac { \left( \mathbf { r } ^ { t } \times \mathbf { r } ^ { tt } \right) \cdot \mathbf { r } ^ { ttt } } { \left| \mathbf { r } ^ { t } \times \mathbf { r } ^ { tt } \right| ^ { 2 } }$$
Find the torsion of the curve defined by $\mathbf { r } ( t ) = \cos 5 t \mathbf { i } + \sin 5 t \mathbf { j } + 4 t \mathbf { k }$.
A) $\frac { 21 } { 41 }$
B) $\frac { 9 } { 41 }$
C) $\frac { 20 } { 41 }$
D) $\frac { 80 } { 41 }$

Question

Quizplus · Accepted Answer

First, we find the derivatives of $\mathbf{r}(t)$:
\begin{align*}
\mathbf { r } ( t ) &= \cos 5 t \mathbf { i } + \sin 5 t \mathbf { j } + 4 t \mathbf { k } \
\mathbf { r } ^ { t } ( t ) &= -5 \sin 5 t \mathbf { i } + 5 \cos 5 t \mathbf { j } + 4 \mathbf { k } \
\mathbf { r } ^ { tt } ( t ) &= -25 \cos 5 t \mathbf { i } - 25 \sin 5 t \mathbf { j } \
\mathbf { r } ^ { ttt } ( t ) &= 125 \sin 5 t \mathbf { i } - 125 \cos 5 t \mathbf { j }
\end{align*}
Then we can calculate the torsion using the given formula:
\begin{align*}
	au &= \frac { \left( \mathbf { r } ^ { t } 	imes \mathbf { r } ^ { tt } ight) \cdot \mathbf { r } ^ { ttt } } { \left| \mathbf { r } ^ { t } 	imes \mathbf { r } ^ { tt } ight| ^ { 2 } } \
&= \frac { \left| \begin{array} { c c c } { \mathbf { i } } & { \mathbf { j } } & { \mathbf { k } } \ { -5 \sin 5 t } & { 5 \cos 5 t } & { 4 } \ { -25 \cos 5 t } & { -25 \sin 5 t } & { 0 } \end{array} ight| \cdot \left( 125 \sin 5 t \mathbf { i } - 125 \cos 5 t \mathbf { j } ight) } { \left| \begin{array} { c c c } { -5 \sin 5 t } & { 5 \cos 5 t } & { 4 } \ { -25 \cos 5 t } & { -25 \sin 5 t } & { 0 } \ { 0 } & { 0 } & { 0 } \end{array} ight| ^ { 2 } } \
&= \frac{500\cos 5t}{\left(\sqrt{625\sin^2 5t+16}ight)^3} \
&= \frac{20}{\sqrt{41}\left(\sqrt{1+ \frac{16}{625}\sin^2 5t}ight)^3}
\end{align*}
Evaluating $	au$ at $t=0$ gives:
\begin{align*}
	au(0) &= \frac{20}{\sqrt{41}\left(\sqrt{1+ \frac{16}{625}(0)}ight)^3} \
&= \frac{20}{\sqrt{41}} \
&= \frac{20\sqrt{41}}{41}
\end{align*}
Therefore, the answer is (C) $\frac{20}{41}$.

The Torsion of a Curve Defined By r(t)\mathbf { r } ( t )r(t)

The Torsion of a Curve Defined By $\mathbf { r } ( t )$