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 p

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.

Answer 1

The curve in terms of arc legth is

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

. Given: The function

r (t) = < 2 \cos t, 2 \sin t, t > s = (\sqrt{5 t})

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{5 t})

t = \frac{s}{\sqrt{5 t}}

Substituting this value of t in equation (1), we get

r (s) = ⟨ 2 \cos (\frac{s}{\sqrt{5 t}}), 2 \sin (\frac{s}{\sqrt{5 t}}), \frac{s}{\sqrt{5 t}} ⟩

Or

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

. Thus, 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

.

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 p

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