To calculate: The coordinates of the point on the helix for arc lengths \(s =\ \sqrt{5}\ and\ s = 4\)
For \(s =\ \sqrt{5},\ \text{the coordinates are}\ \underline{1.081\ 1.683\ 1}\)
For \(s = 4,\ \text{the coordinates are}\ \underline{-0.433\ 1.953\ 1.789}\)
Given:
The function \(r(t)=\ <\ 2\ \cos\ t,\ 2\ \sin\ t,\ t\ >\).
Calculation:
The curve in terms of arc length is,
\(r(s)=2\ \cos\ \left(\frac{s}{\sqrt{5}}\right)i\ +\ 2\ \sin\ \left(\frac{s}{\sqrt{5}}\right)j\ +\ \frac{s}{\sqrt{5}}k\)
Case 1: \(s = \sqrt{5}\)

\(r(\sqrt{5})=2\ \frac{\cos\ \sqrt{5}}{\sqrt{5}}i\ +\ 2\ \sin\ \left(\frac{\sqrt{5}}{\sqrt{5}}\right)j\ +\ \frac{\sqrt{5}}{\sqrt{5}}kt5k\)

\(=2\ \cos(1)i\ +\ 2\ \sin(1)j\ +\ 1k\)

\(=1.081i\ +\ 1.683j\ +\ 1k\) Case 2: \(s = 4\)

\(r(4)=2\ \cos \left(\frac{4}{\sqrt{5}}\right)i\ +\ 2\ \sin \left(\frac{4}{\sqrt{5}}\right)j\ +\ \frac{4}{\sqrt{5}}k\)

\(=\ -0.433i\ +\ 1.953j\ +\ 1.789k\) Thus, for \(s =\ \sqrt{5},\ \text{the coordinates are} (1.081,\ 1.683,\ 1)\ \text{and for}\ s = 4,\ \text{the coordinates are}\ \underline{-0.433\ 1.953\ 1.789}\)

\(r(\sqrt{5})=2\ \frac{\cos\ \sqrt{5}}{\sqrt{5}}i\ +\ 2\ \sin\ \left(\frac{\sqrt{5}}{\sqrt{5}}\right)j\ +\ \frac{\sqrt{5}}{\sqrt{5}}kt5k\)

\(=2\ \cos(1)i\ +\ 2\ \sin(1)j\ +\ 1k\)

\(=1.081i\ +\ 1.683j\ +\ 1k\) Case 2: \(s = 4\)

\(r(4)=2\ \cos \left(\frac{4}{\sqrt{5}}\right)i\ +\ 2\ \sin \left(\frac{4}{\sqrt{5}}\right)j\ +\ \frac{4}{\sqrt{5}}k\)

\(=\ -0.433i\ +\ 1.953j\ +\ 1.789k\) Thus, for \(s =\ \sqrt{5},\ \text{the coordinates are} (1.081,\ 1.683,\ 1)\ \text{and for}\ s = 4,\ \text{the coordinates are}\ \underline{-0.433\ 1.953\ 1.789}\)