Chapter 14 - Calculus of Vector-Valued Functions - 14.1 Vector-Valued Functions - Exercises - Page 710: 23

$r(t)=\langle \pm\frac{\sqrt2}{2}, \pm\frac{\sqrt2}{2}, 0\rangle$, $r(t)=\langle 0, \pm 1, 0\rangle$

Work Step by Step

Solve for where $z(t)=0$ for $0 \le t < 2\pi$: $sintcos2t=0$ $sint=0$ or $cos2t=0$ $t=0, \pi$ or $2t=\frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \frac{7\pi}{2}$ $t=0, \pi$ or $t=\frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$ Plug values of $t$ into $r(t)$: $r(0)=\langle sin(0), cos(0), sin(0)cos(2*0)\rangle$ $r(0)=\langle 0, 1, 0\rangle$ $r(\pi)=\langle sin(\pi), cos(\pi), sin(\pi)cos(2*\pi)\rangle$ $r(\pi)=\langle 0, -1, 0\rangle$ $r(\frac{\pi}{4})=\langle sin(\frac{\pi}{4}), cos(\frac{\pi}{4}), sin(\frac{\pi}{4})cos(2*\frac{\pi}{4})\rangle$ $r(\frac{\pi}{4})=\langle \frac{\sqrt2}{2}, \frac{\sqrt2}{2}, 0\rangle$ $r(\frac{3\pi}{4})=\langle sin(\frac{3\pi}{4}), cos(\frac{3\pi}{4}), sin(\frac{3\pi}{4})cos(2*\frac{3\pi}{4})\rangle$ $r(\frac{3\pi}{4})=\langle \frac{\sqrt2}{2}, -\frac{\sqrt2}{2}, 0\rangle$ $r(\frac{5\pi}{4})=\langle sin(\frac{5\pi}{4}), cos(\frac{5\pi}{4}), sin(\frac{5\pi}{4})cos(2*\frac{5\pi}{4})\rangle$ $r(\frac{5\pi}{4})=\langle -\frac{\sqrt2}{2}, -\frac{\sqrt2}{2}, 0\rangle$ $r(\frac{7\pi}{4})=\langle sin(\frac{7\pi}{4}), cos(\frac{7\pi}{4}), sin(\frac{7\pi}{4})cos(2*\frac{7\pi}{4})\rangle$ $r(\frac{7\pi}{4})=\langle -\frac{\sqrt2}{2}, \frac{\sqrt2}{2}, 0\rangle$

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