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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 \( \mathbf{v}(t) \).

Write a in the form a = a_T * T + a_N * N at the given value of t without finding T and N. r(t) = (-2t + 5)i + (-3t)j + (-t^2)k, t = -3

Learn how to find the tangential and normal components of acceleration, given the position vector r(t) = (-2t + 5)i + (-3t)j + (-t^2)k and t = -3. This problem covers key vector calculus concepts, including velocity, acceleration, and vector decomposition.

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