Vector Functions — Question 9

Question 9 – Solution

Strategy Use the cylinder center and radius for the circular components, then choose a linear height matching the pitch.

Construction Counterclockwise motion from the rightmost point is $$x=2+3\cos t,\qquad y=-1+3\sin t.$$ One revolution is $\Delta t=2\pi$ and should raise $z$ by 4, so $z=(2/\pi)t$. Two revolutions require $0\le t\le 4\pi$. Thus $$\boxed{\vec r(t)=\left\langle 2+3\cos t,-1+3\sin t,2t/\pi\right\rangle},\quad 0\le t\le 4\pi.$$

Verification The cylinder equation is identically 9, and the final height is 8.