Step by step solution to "Verify that ‖rt(s)‖=1.Consider the helix represented investigation by..."

Annette Arroyo

Answered question

2020-10-28

Verify that

‖ r^{t} (s) ‖ = 1

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

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

Talisha

Skilled2020-10-29Added 93 answers

Given:The function

\underset{―}{r (t) = < 2 \cos t, 2 \sin t, t >}

Proofs: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

.On differenting the vector-value function r(s), we get

r^{'} (s) = - \frac{2}{\sqrt{5}} \sin (\frac{s}{\sqrt{5}}) i + \frac{2}{\sqrt{5}} \cos (\frac{s}{\sqrt{5}}) j + \frac{1}{\sqrt{5}} k

From this calcute

‖ r^{'} (s) ‖

‖ r^{'} (s) ‖ = \sqrt{{(- \frac{2}{\sqrt{5}} \sin (\frac{s}{\sqrt{5}}))}^{2} + {(\frac{2}{\sqrt{5}} \cos (\frac{s}{\sqrt{5}}))}^{2} + {(\frac{1}{\sqrt{5}})}^{2}}

= \sqrt{\frac{4}{5} + \frac{1}{5}}

= \sqrt{\frac{5}{5}}

= 1

Hence, it is proved that

‖ r^{'} (s) ‖ = 1

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