Find the curvature of the vector valued function: $$\vec { r } ( t ) = \left\langle e ^ { t } , e ^ { 2 t } , e ^ { 3 t } \right\rangle$$
A) $\kappa ( t ) = \frac { 1 } { 2 }$
B) $\kappa ( t ) = \sqrt { 36 e ^ { 10 t } + 36 e ^ { 8 t } + 4 e ^ { 6 t } }$
C) $\kappa ( t ) = \frac { 2 \sqrt { 9 e ^ { 4 t } + 9 e ^ { 2 t } + 1 } } { \left( 1 + 4 e ^ { 2 t } + 9 e ^ { 4 t } \right) ^ { \frac { 3 } { 2 } } }$
D) $\kappa ( t ) = \sqrt { 9 e ^ { 4 t } + 9 e ^ { 4 t } + 1 }$

Question

Quizplus · Accepted Answer

The curvature $\kappa(t)$ of a vector-valued function $\vec{r}(t)$ is given by $\kappa(t) = \frac{|\vec{r}'(t) \times \vec{r}''(t)|}{|\vec{r}'(t)|^3}$. Calculating the derivatives and their cross product, and then simplifying, yields the expression in choice C.

Find the Curvature of the Vector Valued Function r⃗(t)=⟨et,e2t,e3t⟩\vec { r } ( t ) = \left\langle e ^ { t } , e ^ { 2 t } , e ^ { 3 t } \right\rangler(t)=⟨et,e2t,e3t⟩

Find the Curvature of the Vector Valued Function $\vec { r } ( t ) = \left\langle e ^ { t } , e ^ { 2 t } , e ^ { 3 t } \right\rangle$