Find the distance traveled by a particle moving on the curve given by $\vec { r } ( t ) = \langle \sin ( 3 t ) , \cos ( 3 t ) \rangle$ find the displacement vector as $t$ goes from $t = 0$ to $t = \pi$
A) $\pi \sqrt { 3 }$
B) $3 \pi$
C) $2$
D) $2 \pi$

Question

Quizplus · Accepted Answer

The curve $\vec{r}(t) = \langle \sin(3t), \cos(3t) \rangle$ represents a circle with radius 1, traced 3 times as $t$ goes from 0 to $\pi$. The circumference of one circle is $2\pi$, so the distance traveled is $3 \times 2\pi = 6\pi$. However, the displacement vector is the difference between the final and initial positions, which is $\langle \sin(3\pi), \cos(3\pi) \rangle - \langle \sin(0), \cos(0) \rangle = \langle 0, -1 \rangle - \langle 0, 1 \rangle = \langle 0, -2 \rangle$, with magnitude 2.

Find the Distance Traveled by a Particle Moving on the Curve