The position vector $\mathbf{r}(t)$ for the particle is given by:

$\mathbf{r}(t) = (5.80 \, \text{m})\hat{i} + \left[(2.30 \, \text{m/s})\hat{i} + (3.20 \, \text{m/s})\hat{j}\right] t - (6.40 \, \text{m/s}^2) t^2 \hat{j}$

To find the position, velocity, and acceleration vectors at $t = 4.17 \, \text{s}$, we will perform the following steps:

(a) Position vector $\mathbf{r}(t = 4.17 \, \text{s})$

The general form for the position vector is:

$\mathbf{r}(t) = (5.80 \, \text{m})\hat{i} + (2.30 t) \, \hat{i} + (3.20 t - 6.40 t^2) \, \hat{j}$

At $t = 4.17 \, \text{s}$, substitute this value into the position equation.

$\mathbf{r}(4.17) = (5.80 \, \text{m})\hat{i} + (2.30 \cdot 4.17) \, \hat{i} + \left[3.20 \cdot 4.17 - 6.40 \cdot (4.17)^2\right] \, \hat{j}$

Let's calculate this.

(b) Velocity vector $\mathbf{v}(t = 4.17 \, \text{s})$

To find the velocity, differentiate $\mathbf{r}(t)$ with respect to time $t$.

$\mathbf{v}(t) = \frac{d}{dt} \mathbf{r}(t) = \frac{d}{dt} \left[ (5.80 \, \text{m}) \hat{i} + (2.30 t) \hat{i} + \left(3.20 t - 6.40 t^2 \right) \hat{j} \right]$

$\mathbf{v}(t) = 0 \cdot \hat{i} + (2.30) \hat{i} + (3.20 - 12.80 t) \hat{j}$

At $t = 4.17 \, \text{s}$:

$\mathbf{v}(4.17) = 2.30 \, \hat{i} + \left( 3.20 - 12.80 \cdot 4.17 \right) \hat{j}$

(c) Acceleration vector $\mathbf{a}(t = 4.17 \, \text{s})$

To find the acceleration, differentiate $\mathbf{v}(t)$ with respect to $t$:

$\mathbf{a}(t) = \frac{d}{dt} \mathbf{v}(t) = 0 \cdot \hat{i} + (-12.80) \hat{j}$

The acceleration is constant and equal to:

$\mathbf{a}(t) = -12.80 \, \hat{j} \, \text{m/s}^2$

At $t = 4.17 \, \text{s}$, the acceleration vector remains the same:

$\mathbf{a}(4.17) = -12.80 \, \hat{j} \, \text{m/s}^2$

Let's compute the final expressions for the position and velocity.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to walk through the calculation steps manually, or perhaps guide you through solving it yourself?

Here are 5 related questions to explore the topic further:

1. How do you find the velocity of a particle in two dimensions from a position function?
2. What is the significance of the acceleration vector in uniform motion?
3. How does changing the initial conditions affect the trajectory of the particle?
4. How would the position and velocity vectors differ in three-dimensional motion?
5. Can you explain how differentiation of vector functions applies to physics?

Tip: In motion problems, always check the units of each term to ensure the equation's consistency and correctness.