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The given vector function \( \mathbf{r}(t) = \left\langle e^t \sin t, e^t \cos t, 15e^t \right\rangle \) represents the trajectory of a moving object. We are tasked with finding the **tangential** and **normal** components of the acceleration. Here’s a step-by-step outline of how to approach this:

### 1. Tangential and Normal Components of Acceleration
For any given vector function \( \mathbf{r}(t) \), the acceleration vector \( \mathbf{a}(t) \) can be broken down into:
- **Tangential component \( a_T \)**: This component is along the direction of the velocity vector.
- **Normal component \( a_N \)**: This component is perpendicular to the velocity vector and describes the change in direction.

The formulas for these components are:
- \( a_T = \frac{\mathbf{v}(t) \cdot \mathbf{a}(t)}{|\mathbf{v}(t)|} \)
- \( a_N = \frac{|\mathbf{v}(t) \times \mathbf{a}(t)|}{|\mathbf{v}(t)|} \)

Where \( \mathbf{v}(t) = \mathbf{r}'(t) \) is the velocity vector and \( \mathbf{a}(t) = \mathbf{v}'(t) \) is the acceleration vector.

### 2. Compute the Velocity Vector \( \mathbf{v}(t) = \mathbf{r}'(t) \)
Differentiate \( \mathbf{r}(t) = \left\langle e^t \sin t, e^t \cos t, 15e^t \right\rangle \) with respect to \( t \).

Let me compute this step first.The velocity vector \( \mathbf{v}(t) = \mathbf{r}'(t) \) is:

\[
\mathbf{v}(t) = \left\langle e^t (\sin t + \cos t), e^t (\cos t - \sin t), 15e^t \right\rangle
\]

Next, let's compute the acceleration vector \( \mathbf{a}(t) = \mathbf{v}'(t) \).The acceleration vector \( \mathbf{a}(t) = \mathbf{v}'(t) \) is:

\[
\mathbf{a}(t) = \left\langle 2e^t \cos t, -2e^t \sin t, 15e^t \right\rangle
\]

Now, let's calculate the tangential and normal components of the acceleration using the formulas:

- **Tangential component**: \( a_T = \frac{\mathbf{v}(t) \cdot \mathbf{a}(t)}{|\mathbf{v}(t)|} \)
- **Normal component**: \( a_N = \frac{|\mathbf{v}(t) \times \mathbf{a}(t)|}{|\mathbf{v}(t)|} \)

I will compute these now.The tangential component of the acceleration \( a_T \) is:

\[
a_T = \sqrt{227} e^t
\]

Now, I'll compute the normal component \( a_N \), which involves the cross product \( \mathbf{v}(t) \times \mathbf{a}(t) \).It seems like I can't do more advanced data analysis right now. Please try again later.

However, we successfully computed the tangential component of the acceleration \( a_T = \sqrt{227} e^t \). If you'd like, I can walk you through the remaining steps for computing the normal component manually, or you can try again later when I'm able to run calculations.

Do you have any questions or want further details about the process? Here are five related questions that could expand your understanding:

1. How are the tangential and normal components used in understanding motion?
2. What is the physical meaning of the cross product \( \mathbf{v}(t) \times \mathbf{a}(t) \)?
3. How would the acceleration components change if the trajectory had different exponents?
4. Can you calculate curvature from the normal and tangential components?
5. How do velocity and acceleration relate in non-linear motion?

**Tip:** The tangential acceleration component tells you how fast the speed of an object is changing, while the normal component relates to how sharply it's turning or changing direction.

Consider the following trajectory of a moving object. Find the tangential and normal components of the acceleration. r(t) = <e^t sin t, e^t cos t, 15e^t>

Explore how to determine the tangential and normal components of acceleration for a given vector trajectory r(t) = <e^t sin t, e^t cos t, 15e^t>. The guide includes formulas for velocity and acceleration vectors and step-by-step calculations for the tangential and normal components.

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