Vector Functions — Question 3

Question 3

A particle follows $$\mathbf r(t)=\left\langle 2\cos t,\ 2\sin t,\ t/\pi\right\rangle,\qquad 0\le t\le 4\pi.$$ Without calculus:

1. Describe the trace geometrically and state its axis and radius.

2. Find every point where the particle meets the plane $z=1$.

3. Determine the vertical rise during one complete revolution and the total number of revolutions.

Question 3 – Solution

Strategy. Use the identity $\cos^2t+\sin^2t=1$ and compare the angular parameter with the height.

Step 1: Cylindrical constraint $$x^2+y^2=4\cos^2t+4\sin^2t=4.$$ Thus the trace lies on the cylinder of radius $2$ about the $z$-axis. Since $z=t/\pi$ steadily increases while $(x,y)$ rotates counterclockwise, the curve is a circular helix.

Step 2: Plane intersection Setting $z=1$ gives $$\frac{t}{\pi}=1\quad\Longrightarrow\quad t=\pi.$$ This value is in the interval, and therefore the only point is $$\boxed{\mathbf r(\pi)=\left\langle -2,0,1\right\rangle}.$$

Step 3: Pitch and turns One revolution changes $t$ by $2\pi$, so $$\Delta z=\frac{t+2\pi}{\pi}-\frac{t}{\pi}=2.$$ The interval length $4\pi$ contains $4\pi/(2\pi)=2$ revolutions. Hence the rise per turn is $\boxed{2}$ and the curve makes $\boxed{2}$ complete turns.