Question

In Exercises 28–30 show that the given vector-valued functions are not arc length parametrizations. Then find an arc length parametrization for the curves defined by those functions.

$\mathbf{r}\left(t\right)=⟨3+2t,4-t,-1+5t⟩$

Short answer

The arc length parametrization is.

Step-by-step solution

Step 1. Given information.

The given function is$\mathbf{r}\left(t\right)=⟨3+2t,4-t,-1+5t⟩$

Step 2. Required answer.

Now,

$$\begin{gathered} ||r^{\text{'}}\left(t\right)||=\sqrt{(2{)}^2+(-1{)}^2+(5{)}^2}=\sqrt{30} \\ \text{Consider}{\left.{\int }_d^c||r^{\text{'}}\left(t\right)||dt={\int }_d^c\sqrt{30}dt=\sqrt{30}t\right]}_c^d \\ =\sqrt{30}(d-c)\ne d-c \end{gathered}$$

Hence the given vector-valued functions are not arc length parametrizations on the interval $[0,1]$the arc length of the curve is $\sqrt{30}$

Now, $\text{replace t with}\frac{5}{\sqrt{30}}\text{in}r\left(t\right)$,we get,