Show that if $\mathbf { r } ^ { \prime } ( t )$ and $\mathbf { r } ^ { \prime \prime } ( t )$ are parallel at some point on the curve described by $\mathbf { r } ( t )$ , then the curvature at that point is 0. Give an example of a curve $\mathbf { r } ( t )$ for which $\mathbf { r } ^ { \prime } ( t )$ and $\mathbf { r } ^ { \prime \prime } ( t )$ are always parallel.

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The Answer of Show that if $\mathbf { r } ^ { \prime } ( t )$ and $\mathbf { r } ^ { \prime \prime } ( t )$ are parallel at some point on the curve described by $\mathbf { r } ( t )$ , then the curvature at that point is 0. Give an example of a curve $\mathbf { r } ( t )$ for which $\mathbf { r } ^ { \prime } ( t )$ and $\mathbf { r } ^ { \prime \prime } ( t )$ are always parallel.

Show That If r′(t)\mathbf { r } ^ { \prime } ( t )r′(t)

Show That If $\mathbf { r } ^ { \prime } ( t )$