Consider the helix represented investigation by the vector-valued function r(t)= < 2 cos t, 2 sin t, t >. 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.

Question
Consider the helix represented investigation by the vector-valued function \(r(t)=\ <\ 2\ \cos\ t,\ 2\ \sin\ t,\ t\ >\). 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.

Answers (1)

2020-10-29
The curve in terms of arc legth is \(r(s)=2\ \cos\ \left(\frac{s}{\sqrt{5}}\right)i\ +\ 2\ \sin\ \left(\frac{s}{\sqrt{5}}\right)j\ +\ \frac{s}{\sqrt{5}}k\). Given: The function \(r(t)=\ <\ 2\ \cos\ t,\ 2\ \sin\ t,\ t\ >\ s=(\sqrt{5t})\) Calculate: The given vector-function for the path is \(r(t)=\ <\ 2\ \cos\ t,\ 2\ \sin\ t,\ t\ >\)........(1) The length of the curve is \(s=(\sqrt{5t})\)
\(t=\ \frac{s}{\sqrt{5t}}\) Substituting this value of t in equation (1), we get \(r(s)=\ \langle2\ \cos\ \left(\frac{s}{\sqrt{5t}}\right),\ 2\ \sin\ \left(\frac{s}{\sqrt{5t}}\right),\ \frac{s}{\sqrt{5t}}\rangle\) Or \(r(s)=2\ \cos\ \left(\frac{s}{\sqrt{5}}\right)i\ +\ 2\ \sin\ \left(\frac{s}{\sqrt{5}}\right)j\ +\ \frac{s}{\sqrt{5}}k\). Thus, the curve in terms of arc length is \(r(s)=2\ \cos\ \left(\frac{s}{\sqrt{5}}\right)i\ +\ 2\ \sin\ \left(\frac{s}{\sqrt{5}}\right)j\ +\ \frac{s}{\sqrt{5}}k\).
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