sanuluy

2020-10-28

Consider the helix represented investigation by the vector-valued function
$r(t)=\text{}\text{}2\text{}\mathrm{cos}\text{}t,\text{}2\text{}\mathrm{sin}\text{}t,\text{}t\text{}$ . Solve for t in the relationship derived in part (a), and substitute the result into the
original set of parametric equations. This yields a parametrization of the curve in terms of
the arc length parameter s.

Brittany Patton

Skilled2020-10-29Added 100 answers

The curve in terms of arc legth is $r(s)=2\text{}\mathrm{cos}\text{}\left(\frac{s}{\sqrt{5}}\right)i\text{}+\text{}2\text{}\mathrm{sin}\text{}\left(\frac{s}{\sqrt{5}}\right)j\text{}+\text{}\frac{s}{\sqrt{5}}k$ .
Given:
The function $r(t)=\text{}\text{}2\text{}\mathrm{cos}\text{}t,\text{}2\text{}\mathrm{sin}\text{}t,\text{}t\text{}\text{}s=(\sqrt{5t})$
Calculate:
The given vector-function for the path is
$r(t)=\text{}\text{}2\text{}\mathrm{cos}\text{}t,\text{}2\text{}\mathrm{sin}\text{}t,\text{}t\text{}$ ........(1)
The length of the curve is
$s=(\sqrt{5t})$

$t=\text{}\frac{s}{\sqrt{5t}}$
Substituting this value of t in equation (1), we get
$r(s)=\text{}\u27e82\text{}\mathrm{cos}\text{}\left(\frac{s}{\sqrt{5t}}\right),\text{}2\text{}\mathrm{sin}\text{}\left(\frac{s}{\sqrt{5t}}\right),\text{}\frac{s}{\sqrt{5t}}\u27e9$
Or
$r(s)=2\text{}\mathrm{cos}\text{}\left(\frac{s}{\sqrt{5}}\right)i\text{}+\text{}2\text{}\mathrm{sin}\text{}\left(\frac{s}{\sqrt{5}}\right)j\text{}+\text{}\frac{s}{\sqrt{5}}k$ .
Thus, the curve in terms of arc length is $r(s)=2\text{}\mathrm{cos}\text{}\left(\frac{s}{\sqrt{5}}\right)i\text{}+\text{}2\text{}\mathrm{sin}\text{}\left(\frac{s}{\sqrt{5}}\right)j\text{}+\text{}\frac{s}{\sqrt{5}}k$ .

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