The problem asks to write the acceleration vector a in the form $a = a_T \mathbf{T} + a_N \mathbf{N}$, where $a_T$ and $a_N$ are the tangential and normal components of the acceleration, and $\mathbf{T}$ and $\mathbf{N}$ are the unit tangent and normal vectors, respectively.

You are given the position vector:

$\mathbf{r}(t) = (-2t + 5)\mathbf{i} + (-3t)\mathbf{j} + (-t^2)\mathbf{k}$ and $t = -3$.

Steps to Solve:

1. Velocity Vector $\mathbf{v}(t)$: Take the derivative of $\mathbf{r}(t)$ with respect to $t$ to find the velocity vector.

2. Acceleration Vector $\mathbf{a}(t)$: Take the derivative of $\mathbf{v}(t)$ with respect to $t$ to find the acceleration vector.

3. Magnitude of the Velocity $|\mathbf{v}(t)|$: Compute the magnitude of $\mathbf{v}(t)$ at $t = -3$.

4. Tangential Component $a_T$: The tangential component of the acceleration is given by: $a_T = \frac{\mathbf{v}(t) \cdot \mathbf{a}(t)}{|\mathbf{v}(t)|}$ at $t = -3$.

5. Normal Component $a_N$: The normal component of the acceleration is given by: $a_N = \sqrt{|\mathbf{a}(t)|^2 - a_T^2}$

Let me compute these values.Here are the computed values:

1. Velocity vector at $t = -3$: $\mathbf{v}(-3) = \langle -2, -3, 6 \rangle$

2. Acceleration vector at $t = -3$: $\mathbf{a}(-3) = \langle 0, 0, -2 \rangle$

3. Magnitude of velocity $|\mathbf{v}(-3)|$: $|\mathbf{v}(-3)| = 7$

4. Tangential component of acceleration $a_T$: $a_T = -1.714$

5. Normal component of acceleration $a_N$: $a_N = 1.03$

Thus, the acceleration vector in the form $a = a_T \mathbf{T} + a_N \mathbf{N}$ is:

$a = -1.714 \mathbf{T} + 1.03 \mathbf{N}$

Let me know if you'd like further details or have any questions!

Here are 5 related questions:

1. How is the normal vector $\mathbf{N}$ related to the curvature of the path?
2. Can the tangential component $a_T$ be zero? If so, under what conditions?
3. How would you calculate the curvature $\kappa$ of the curve given the position vector $\mathbf{r}(t)$?
4. What physical meaning does the magnitude of the velocity vector represent in this context?
5. How does the direction of the velocity vector relate to the tangent vector $\mathbf{T}$?

Tip: To find the unit tangent vector $\mathbf{T}$, normalize the velocity vector $\mathbf{v}(t)$.