Find the curvature of the space curve.
-$$r ( t ) = - 5 i + ( 4 + 2 t ) j + \left( t ^ { 2 } + 8 \right) \mathbf { k }$$
A) $\kappa = \frac { 1 } { 2 \left( t ^ { 2 } + 1 \right) ^ { 3 / 2 } }$
B) $\kappa = - \frac { 1 } { 2 \left( t ^ { 2 } + 1 \right) ^ { 3 / 2 } }$
C) $\kappa = \frac { 1 } { 2 \sqrt { t ^ { 2 } + 1 } }$
D) $\kappa = \frac { 1 } { \left( t ^ { 2 } + 1 \right) ^ { 3 / 2 } }$

Question

Quizplus · Accepted Answer

The curvature $\kappa$ of a space curve defined by a vector function $r(t)$ is given by $\kappa = \frac{\|r'(t) \times r''(t)\|}{\|r'(t)\|^3}$. For $r(t) = -5i + (4 + 2t)j + (t^2 + 8)k$, we first find the derivatives $r'(t)$ and $r''(t)$, then compute the cross product and norms as required. The correct formula for curvature, given the specific form of $r(t)$, is $\kappa = \frac { 1 } { 2 \left( t ^ { 2 } + 1 \right) ^ { 3 / 2 } }$, which matches choice A.

Find the Curvature of the Space Curve r(t)=−5i+(4+2t)j+(t2+8)kr ( t ) = - 5 i + ( 4 + 2 t ) j + \left( t ^ { 2 } + 8 \right) \mathbf { k }r(t)=−5i+(4+2t)j+(t2+8)k

Find the Curvature of the Space Curve $r ( t ) = - 5 i + ( 4 + 2 t ) j + \left( t ^ { 2 } + 8 \right) \mathbf { k }$