Question

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

Solution

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Answered 12 months ago
Answered 12 months ago
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The statement is true. Let's see why.

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

For every tt we have that

T(t)=r(t)r(t)=r(t)c\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

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

and therefore

N(t)=T(t)T(t)=r(t)cr(t)c=r(t)cr(t)c=r(t)ccr(t)=r(t)r(t).\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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