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 $\lvert\mathbf{r}'(t)\rvert=c$, 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

Verified
Step 1
1 of 3

The 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

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

## Recommended textbook solutions

#### Thomas' Calculus

14th EditionISBN: 9780134438986Christopher E Heil, Joel R. Hass, Maurice D. Weir
10,144 solutions

#### Calculus: Early Transcendentals

1st EditionISBN: 9780534465544Tan, Soo
8,135 solutions

#### Calculus: Early Transcendentals

8th EditionISBN: 9781285741550 (2 more)James Stewart
11,085 solutions

#### Calculus: Early Transcendentals

9th EditionISBN: 9781337613927 (2 more)Daniel K. Clegg, James Stewart, Saleem Watson
11,050 solutions