# Find the coordinates of the point on the helix for arc lengths s = sqrt{5} and s = 4. Consider the helix represented investigation by the vector-valued function r(t)= < 2 cos t, 2 sin t, t >.

Question
Find the coordinates of the point on the helix for arc lengths $$s =\ \sqrt{5}\ and\ s = 4$$. Consider the helix represented investigation by the vector-valued function $$r(t)=\ <\ 2\ \cos\ t,\ 2\ \sin\ t,\ t\ >$$.

2020-11-24
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 $$s =\ \sqrt{5},\ \text{the coordinates are} (1.081,\ 1.683,\ 1)\ \text{and for}\ s = 4,\ \text{the coordinates are}\ \underline{-0.433\ 1.953\ 1.789}$$

### Relevant Questions

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