For $\vec { r } ( t ) = \langle \sin ( 3 t ) , \cos ( 3 t ) , t \rangle$ find the displacement vector as $t$ goes from $t = 0$ to $t = \pi$
A) $\langle - 2,0 , \pi \rangle$
B) $\langle - 1,0 , \pi \rangle$
C) $\langle 1,0 , \pi \rangle$
D) $\langle 1,1 , \pi \rangle$

Question

Quizplus · Accepted Answer

The displacement vector is found by evaluating $\vec{r}(\pi) - \vec{r}(0)$. For $\vec{r}(\pi)$, we get $\langle \sin(3\pi), \cos(3\pi), \pi \rangle = \langle 0, -1, \pi \rangle$. For $\vec{r}(0)$, we get $\langle \sin(0), \cos(0), 0 \rangle = \langle 0, 1, 0 \rangle$. Subtracting these gives $\langle 0 - 0, -1 - 1, \pi - 0 \rangle = \langle 0, -2, \pi \rangle$, which simplifies to $\langle -2, 0, \pi \rangle$.

For r⃗(t)=⟨sin⁡(3t),cos⁡(3t),t⟩\vec { r } ( t ) = \langle \sin ( 3 t ) , \cos ( 3 t ) , t \rangler(t)=⟨sin(3t),cos(3t),t⟩

For $\vec { r } ( t ) = \langle \sin ( 3 t ) , \cos ( 3 t ) , t \rangle$