Vector Functions — Question 5

Question 5 – Solution

Strategy. Parametrize the cylinder first; then use the plane equation to determine the remaining component.

Step 1: Parametrize the circle The cylinder equation suggests $$x=3\cos t,\qquad y=3\sin t,$$ because these expressions satisfy $x^2+y^2=9$ identically.

Step 2: Impose the plane Substitute the chosen $x$ and $y$ into $z=x-2y$: $$z=3\cos t-6\sin t.$$ Therefore $$\boxed{\mathbf r(t)=\left\langle 3\cos t,\ 3\sin t,\ 3\cos t-6\sin t\right\rangle}, \qquad \boxed{0\le t<2\pi}.$$

Step 3: Initial conditions At $t=0$, $$\mathbf r(0)=\left\langle 3,0,3\right\rangle.$$ For small positive $t$, $\sin t>0$, so $y=3\sin t>0$; the required initial direction is satisfied.

Verification. Squaring the first two components gives $9(\cos^2t+\sin^2t)=9$, and the third component is exactly the first minus twice the second. The half-open interval avoids repeating the starting point.