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# The position vector r describes the path of an object moving in space. (a) Find the velocity vector, speed, and acceleration vector of the object. Position Vector: r(t) = 3ti + tj + ¼t²k Time: t = 2

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If $f,\,g$ and $h$ are twice-differentiable functions of $t$ and $\mathbf{r}$ is a vector-valued function given by $\mathbf{r}(t)=f(t)\mathbf{i}+g(t)\mathbf{j}+h(t)\mathbf{k}$, then the velocity vector at time $t$ is:

\begin{align} \mathbf{v}(t)=\mathbf{r}'(t)=f'(t)\mathbf{i}+g'(t)\mathbf{j}+h'(t)\mathbf{k}. \end{align}

The acceleration vector at time $t$ is defined as:

\begin{align} \mathbf{a}(t)=\mathbf{r}''(t)=f''(t)\mathbf{i}+g''(t)\mathbf{j}+h''(t)\mathbf{k}, \end{align}

and the speed at time $t$ is defined as the magnitude of the velocity vector $\mathbf{v}$:

\begin{align} ||\mathbf{v}(t)||=||\mathbf{r}'(t)||=\sqrt{\left[f'(t)\right]^2+\left[g'(t)\right]^2+\left[h'(t)\right]^2}. \end{align}

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