Annette Arroyo

2020-10-28

Verify that $\Vert {r}^{t}(s)\Vert =1$ .Consider the helix represented investigation by the vector-valued function
$r(t)=\text{}\text{}2\text{}\mathrm{cos}\text{}t,\text{}2\text{}\mathrm{sin}\text{}t,\text{}t\text{}$ .

Talisha

Skilled2020-10-29Added 93 answers

Given:The function $\underset{\u2015}{r(t)=\text{}\text{}2\text{}\mathrm{cos}\text{}t,\text{}2\text{}\mathrm{sin}\text{}t,\text{}t\text{}}$ Proofs:The curve in terms of arc length is,$r(s)=2\text{}\mathrm{cos}\text{}\left(\frac{s}{\sqrt{5}}\right)i\text{}+\text{}2\text{}\mathrm{sin}\text{}\left(\frac{s}{\sqrt{5}}\right)j\text{}+\text{}\frac{s}{\sqrt{5}}k$ .On differenting the vector-value function r(s), we get${r}^{\prime}(s)=\text{}-\frac{2}{\sqrt{5}}\text{}\mathrm{sin}\left(\frac{s}{\sqrt{5}}\right)i\text{}+\text{}\frac{2}{\sqrt{5}}\text{}\mathrm{cos}\left(\frac{s}{\sqrt{5}}\right)j\text{}+\text{}\frac{1}{\sqrt{5}}k$ From this calcute $\Vert {r}^{\prime}(s)\Vert $ as$\Vert {r}^{\prime}(s)\Vert =\text{}\sqrt{{(-\frac{2}{\sqrt{5}}\text{}\mathrm{sin}\left(\frac{s}{\sqrt{5}}\right))}^{2}\text{}+\text{}{(\frac{2}{\sqrt{5}}\text{}\mathrm{cos}\left(\frac{s}{\sqrt{5}}\right))}^{2}\text{}+\text{}{\left(\frac{1}{\sqrt{5}}\right)}^{2}}$

$=\text{}\sqrt{\frac{4}{5}\text{}+\text{}\frac{1}{5}}$

$=\text{}\sqrt{\frac{5}{5}}$

$=1$ Hence, it is proved that $\Vert {r}^{\prime}(s)\Vert =1$

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