Cylindrical Coordinates — Question 9

Question 9 – Solution

Strategy Substitute into $x=r\cos\theta$, $y=r\sin\theta$ and examine the change when $t$ increases by $2\pi$.

Rectangular form $$\boxed{\vec r(t)=\left\langle 2\cos t,2\sin t,3t\right\rangle}.$$ It is a circular helix of radius $2$ around the $z$-axis.

Pitch One complete revolution corresponds to $\Delta t=2\pi$. Therefore $$\boxed{\Delta z=3(2\pi)=6\pi}.$$ The constant $r$ keeps the curve on a cylinder, while the linear $z$-coordinate produces uniform vertical advance.