To find the tangential ($a_T$) and normal ($a_N$) components of the acceleration vector, we follow these steps:

Given the position vector $\mathbf{r}(t) = t \hat{i} + 2e^t \hat{j} + e^{2t} \hat{k}$, we need to:

1. Differentiate the position vector $\mathbf{r}(t)$ to get the velocity vector $\mathbf{v}(t)$.
2. Differentiate $\mathbf{v}(t)$ to get the acceleration vector $\mathbf{a}(t)$.
3. Use the formulas for the tangential and normal components of acceleration:
   • $a_T = \frac{\mathbf{v}(t) \cdot \mathbf{a}(t)}{|\mathbf{v}(t)|}$
   • $a_N = \frac{| \mathbf{v}(t) \times \mathbf{a}(t) |}{|\mathbf{v}(t)|}$

Let's compute step by step.

Step 1: Velocity vector $\mathbf{v}(t)$

The velocity is the derivative of $\mathbf{r}(t)$: $\mathbf{v}(t) = \frac{d}{dt} \left( t \hat{i} + 2e^t \hat{j} + e^{2t} \hat{k} \right) = \hat{i} + 2e^t \hat{j} + 2e^{2t} \hat{k}$

Step 2: Acceleration vector $\mathbf{a}(t)$

The acceleration is the derivative of $\mathbf{v}(t)$: $\mathbf{a}(t) = \frac{d}{dt} \left( \hat{i} + 2e^t \hat{j} + 2e^{2t} \hat{k} \right) = 2e^t \hat{j} + 4e^{2t} \hat{k}$

Step 3: Tangential component $a_T$

We first compute the dot product of $\mathbf{v}(t)$ and $\mathbf{a}(t)$: $\mathbf{v}(t) \cdot \mathbf{a}(t) = (1)(0) + (2e^t)(2e^t) + (2e^{2t})(4e^{2t}) = 4e^{2t} + 8e^{4t}$

Now, compute the magnitude of the velocity vector $\mathbf{v}(t)$: $|\mathbf{v}(t)| = \sqrt{(1)^2 + (2e^t)^2 + (2e^{2t})^2} = \sqrt{1 + 4e^{2t} + 4e^{4t}}$

Finally, the tangential component is: $a_T = \frac{4e^{2t} + 8e^{4t}}{\sqrt{1 + 4e^{2t} + 4e^{4t}}}$

Step 4: Normal component $a_N$

We need to compute the magnitude of the cross product $\mathbf{v}(t) \times \mathbf{a}(t)$.

Let me calculate these final components for you.The tangential ($a_T$) and normal ($a_N$) components of the acceleration vector are:

$a_T = \frac{(8e^{2t} + 4)e^{2t}}{\sqrt{4e^{4t} + 4e^{2t} + 1}}$

$a_N = \frac{2\sqrt{4e^{6t} + 4e^{4t} + e^{2t}}}{\sqrt{4e^{4t} + 4e^{2t} + 1}}$

These are the required components of the acceleration vector.

Would you like further details on any step or a more detailed explanation?

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Here are five related questions for you to consider:

1. How do the velocity and acceleration vectors behave as $t \to \infty$?
2. What is the geometric interpretation of the tangential and normal components?
3. Can you find the curvature of the trajectory given $\mathbf{r}(t)$?
4. How would the results change if the position vector were in 2D instead of 3D?
5. How can these components help in understanding the motion in physics?

Tip: The tangential component measures the acceleration along the path, while the normal component gives the acceleration perpendicular to the path.