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\newcommand{\un}{\underline}
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\begin{center}
\textbf{Vector Functions and Curves}

\textit{\textbf{One variable functions}}
\end{center}

\textbf{Question}

Find the velocity, speed and acceleration of the particle with
position given by $\un{r}(t)$ at time
$t$. Also determine the particles path.

$$\underline{r}= 3\cos t\underline{i}+ 4\cos t\underline{j}+ 5\sin
t\underline{k}$$

\textbf{Answer}

Position: $\underline{r}=3\cos t\underline{i}+ 4\cos t\underline{j}
+5\sin t \underline{k}$

Velocity: $\underline{v}=-3\sin t \underline{i} -4\sin t \underline{j}
+5\cos t\underline{k}$

Speed: $\it{v}=\sqrt{9\sin^2 t+16\sin^2 t+ 25\cos^2 t}=5$

Acceleration: $\underline{a}=-3\cos t\underline{i} -4\cos t
\underline{j} -5\sin t \underline{k}=-\underline{r}$

Path: the circle of intersection of the sphere $x^2+y^2+z^2=25$ and
the plane $4x=3y$.
\end{document}