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} a n d s = 4

. Consider the helix represented investigation by the vector-valued function

r (t) = < 2 \cos t, 2 \sin t, t >

.

Answer 1

To calculate: The coordinates of the point on the helix for arc lengths

s = \sqrt{5} a n d s = 4

For

s = \sqrt{5}, the coordinates are \underset{―}{1.081 1.683 1}

For

s = 4, the coordinates are \underset{―}{- 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 (\frac{s}{\sqrt{5}}) i + 2 \sin (\frac{s}{\sqrt{5}}) j + \frac{s}{\sqrt{5}} k

Case 1:

s = \sqrt{5}

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

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

= 1.081 i + 1.683 j + 1 k

Case 2:

s = 4

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

= - 0.433 i + 1.953 j + 1.789 k

Thus, for

s = \sqrt{5}, the coordinates are (1.081, 1.683, 1) and for s = 4, the coordinates are \underset{―}{- 0.433 1.953 1.789}

Expert Answer