Let's analyze the problem from the image. It gives the following information:

• The acceleration vector is given as: $\mathbf{a}(t) = \left( -9\cos(3t), -9\sin(3t), 0.5 \right)$

• The initial velocity vector: $\mathbf{v}(0) = (1, 0, 1)$

• The initial position vector: $\mathbf{r}(0) = (1, 1, 1)$

You are asked to compute: A. The velocity vector $\mathbf{v}(t)$ as a function of time $t$. B. The position vector $\mathbf{r}(t)$ as a function of time $t$.

Solution Outline:

1. Find the velocity vector $\mathbf{v}(t)$ by integrating the acceleration vector $\mathbf{a}(t)$.

2. Find the position vector $\mathbf{r}(t)$ by integrating the velocity vector $\mathbf{v}(t)$.

I'll perform the calculations step-by-step and explain each result. Let's first integrate to find the velocity vector.The velocity vector $\mathbf{v}(t)$ is given by integrating the acceleration vector:

$\mathbf{v}(t) = \left( 1 - 3\sin(3t), 3\cos(3t), 0.5t + 1 \right)$

Now, let's proceed to find the position vector $\mathbf{r}(t)$ by integrating the velocity vector.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can guide you through the next steps manually:

Step for the Position Vector:

To find the position vector $\mathbf{r}(t)$, integrate the velocity components:

$r_x(t) = \int (1 - 3\sin(3t)) dt + 1$ $r_y(t) = \int 3\cos(3t) dt + 1$ $r_z(t) = \int (0.5t + 1) dt + 1$

After solving these integrals, you can add the initial position $\mathbf{r}(0) = (1, 1, 1)$.

Would you like further assistance with the integration steps, or do you have any questions?

Here are 5 related questions:

1. How would you compute the total displacement from $t = 0$ to $t = T$?
2. How is the magnitude of the velocity vector $\mathbf{v}(t)$ calculated?
3. Can you describe the physical meaning of each component of the acceleration vector?
4. What role do the initial conditions $\mathbf{v}(0)$ and $\mathbf{r}(0)$ play in these calculations?
5. How would adding another component to the acceleration vector affect the solution?

Tip: Always double-check integration constants by considering the initial conditions to avoid mistakes.