Find the principal unit normal vector N for the curve r(t).
-r(t) = (2t sin t + 2 cos t)i + (2t cos t - 2 sin t))k
A) $\mathbf { N } = ( - \sin t ) \mathbf { i } - ( \cos t ) \mathbf { k }$
B) $\mathbf { N } = ( \cos t ) \mathbf { i } - ( \sin t ) \mathbf { k }$
C) $\mathbf { N } = - \frac { \sqrt { 2 } } { 2 } ( \cos t ) \mathbf { i } - \frac { \sqrt { 2 } } { 2 } ( \sin t ) \mathbf { k }$
D) $\mathbf { N } = ( \sin t ) \mathbf { i } + ( \cos t ) \mathbf { k }$

Question

Find the principal unit normal vector N for the curve r(t).
-r(t) = (2t sin t + 2 cos t)i + (2t cos t - 2 sin t))k 
A) $\mathbf { N } = ( - \sin t ) \mathbf { i } - ( \cos t ) \mathbf { k }$
B) $\mathbf { N } = ( \cos t ) \mathbf { i } - ( \sin t ) \mathbf { k }$
C) $\mathbf { N } = - \frac { \sqrt { 2 } } { 2 } ( \cos t ) \mathbf { i } - \frac { \sqrt { 2 } } { 2 } ( \sin t ) \mathbf { k }$
D) $\mathbf { N } = ( \sin t ) \mathbf { i } + ( \cos t ) \mathbf { k }$

Quizplus · Accepted Answer

The principal unit normal vector $\mathbf{N}$ is found by first calculating the derivative of $\mathbf{r}(t)$ to get the tangent vector $\mathbf{T}(t)$, then differentiating $\mathbf{T}(t)$ and normalizing it. The given curve $\mathbf{r}(t) = (2t \sin t + 2 \cos t)\mathbf{i} + (2t \cos t - 2 \sin t)\mathbf{k}$ leads to a tangent vector that, when normalized and differentiated again, results in the principal unit normal vector $\mathbf{N} = (-\sin t)\mathbf{i} - (\cos t)\mathbf{k}$, after normalization and ensuring it points towards the curve's concavity.

Find the Principal Unit Normal Vector N for the Curve N=(−sin⁡t)i−(cos⁡t)k\mathbf { N } = ( - \sin t ) \mathbf { i } - ( \cos t ) \mathbf { k }N=(−sint)i−(cost)k

Find the Principal Unit Normal Vector N for the Curve $\mathbf { N } = ( - \sin t ) \mathbf { i } - ( \cos t ) \mathbf { k }$