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Question Number 164639 by cortano1 last updated on 20/Jan/22

Answered by mr W last updated on 20/Jan/22

$$x=a\cos 2\pi t\Rightarrow \frac{dx}{dt}=-2\pi a\sin 2\pi t$$$$y=a\sin 2\pi t\Rightarrow \frac{dy}{dt}=2\pi a\cos 2\pi t$$$$z=bt\Rightarrow \frac{dz}{dt}=b$$$$att=1:$$$$x=a,y=0,z=b$$$$\frac{dx}{dt}=0,\frac{dy}{dt}=2\pi a,\frac{dz}{dt}=b$$$$$$$$tangentvector:$$$$(0,2\pi a,b)$$$$$$$$tangentline:$$$$\frac{x-a}{0}=\frac{y-0}{2\pi a}=\frac{z-b}{b}$$$$\Rightarrow \frac{a-x}{0}=\frac{y}{2\pi a}=\frac{z-b}{b}$$$$$$$$speed:$$$$v=\sqrt{{\left(0\right)}^2+{\left(2\pi a\right)}^2+b^2}=\sqrt{4{\pi }^2{a}^2+b^2}$$$$$$$$normalplane:$$$$0(x-a)+2\pi a(y-0)+b(z-b)=0$$$$\Rightarrow 2\pi ay+bz=b^2$$

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