https://file.aiquickdraw.com/mathbot/file/1726465212440p7r2wmaz.jpg

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.

Given that the acceleration vector is a(t) = (-9cos(3t), -9sin(3t), 0.5), the initial velocity is v(0) = (1, 0, 1), and the initial position is r(0) = (1, 1, 1), compute the velocity vector v(t) and the position vector r(t).

Learn how to compute velocity and position vectors from a given acceleration vector using integration. This problem explores key vector calculus concepts and uses the fundamental theorem of calculus to solve the equations. Suitable for advanced high school and college students.

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