Answered: Show that at every point on the curve r…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Show that at every point on the curve
r (t) = <e^t cost , e^t sint , e^t>

the angle between the unit tangent vector and the z-axis is the same. Then
show that the same result holds for the unit normal and binormal vectors

Expert Solution

Step 1

The curve is $r (t) = <e^{t} \cos t, e^{t} \sin t, e^{t}>$ .

Find the derivative $r' (t)$ and the value $|r' (t)|$ .

$\begin{array}{rcl} r' (t) & = & <- e^{t} \sin t + \cos t e^{t}, e^{t} \cos t + e^{t} \sin t, e^{t}> \\ = & <e^{t} (\cos t - \sin t), e^{t} (\cos t + \sin t), e^{t}> \\ |r' (t)| & = & \sqrt{e^{2 t} {(\cos t - \sin t)}^{2} + e^{2 t} {(\cos t + \sin t)}^{2} + e^{2 t}} \\ = & \sqrt{e^{2 t} ({(\cos t - \sin t)}^{2} + {(\cos t + \sin t)}^{2} + 1)} \\ = & \sqrt{e^{2 t} (\cos^{2} t + \sin^{2} t - 2 \cos t \sin t + \cos^{2} t + \sin^{2} t + 2 \cos t \sin t + 1)} \\ = & \sqrt{e^{2 t} (1 + 1 + 1)} \\ = & \sqrt{3 e^{2 t}} \\ = & \sqrt{3} e^{t} \end{array}$

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