## Related questions with answers

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false.

If $\lvert\mathbf{r}'(t)\rvert=c$, where $c$ is a nonzero constant, then the unit normal to the curve $C$ defined by $\mathbf{r}(t)$ is given by $\mathbf{N}=\frac{\mathbf{r}''}{\lvert \mathbf{r}''\rvert}$.

Solution

VerifiedThe statement is true. Let's see why.

Let $\textbf{r}\left(t\right)$ be a vector function such that $\left|\textbf{r}'\left(t\right)\right|=c$ where $c$ is nonzero constant. Let $C$ be the curve defined by $\textbf{r}\left(t\right)$.

For every $t$ we have that

$\textbf{T}\left(t\right)=\displaystyle\frac{\textbf{r}\left(t\right)}{\left|\textbf{r}'\left(t\right)\right|}=\displaystyle\frac{\textbf{r}'\left(t\right)}{c}$

which gives us that

$\textbf{T'}\left(t\right)=\displaystyle\frac{\textbf{r}''\left(t\right)}{c}$

and therefore

$\textbf{N}\left(t\right)=\displaystyle\frac{\textbf{T}'\left(t\right)}{\left|\textbf{T}'\left(t\right)\right|}=\displaystyle\frac{\displaystyle\frac{\textbf{r}''\left(t\right)}{c}}{\left|\displaystyle\frac{\textbf{r}''\left(t\right)}{c}\right|}=\displaystyle\frac{\displaystyle\frac{\textbf{r}''\left(t\right)}{c}}{\displaystyle\frac{\left|\textbf{r}''\left(t\right)\right|}{c}}=\displaystyle\frac{\textbf{r}''\left(t\right)}{c}\displaystyle\frac{c}{\left|\textbf{r}''\left(t\right)\right|}=\displaystyle\frac{\textbf{r}''\left(t\right)}{\left|\textbf{r}''\left(t\right)\right|}.$

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