Write a in the form aequalsa Subscript Upper TTplusa Subscript Upper NN at the given value of t without finding T and N.
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Part 1
Bold a left parenthesis 0 right parenthesis equals left parenthesis nothing right parenthesis Bold Upper T plus left parenthesis nothing right parenthesis Bold Upper N

To write the acceleration vector \(\mathbf{a}(t)\) in the form \(\mathbf{a}(t) = a_T \mathbf{T} + a_N \mathbf{N}\) at \(t = 0\) without finding the tangent vector \(\mathbf{T}\) and the normal vector \(\mathbf{N}\), we first need to determine \(\mathbf{a}(0)\) directly from the given position vector \(\mathbf{r}(t)\).

Given:
\[
\mathbf{r}(t) = \left(7e^t \sin t\right) \mathbf{i} + \left(7e^t \sqrt{3}\right) \mathbf{j} + \left(7e^t \cos t\right) \mathbf{k}
\]

### Step 1: Compute the velocity vector \(\mathbf{v}(t)\)
The velocity vector \(\mathbf{v}(t)\) is the first derivative of the position vector \(\mathbf{r}(t)\) with respect to time \(t\):
\[
\mathbf{v}(t) = \frac{d\mathbf{r}(t)}{dt}
\]

Compute the derivative component-wise:
\[
\mathbf{v}(t) = \left(\frac{d}{dt}\left(7e^t \sin t\right)\right) \mathbf{i} + \left(\frac{d}{dt}\left(7e^t \sqrt{3}\right)\right) \mathbf{j} + \left(\frac{d}{dt}\left(7e^t \cos t\right)\right) \mathbf{k}
\]

For the \(\mathbf{i}\) component:
\[
\frac{d}{dt}\left(7e^t \sin t\right) = 7e^t \sin t + 7e^t \cos t = 7e^t (\sin t + \cos t)
\]

For the \(\mathbf{j}\) component:
\[
\frac{d}{dt}\left(7e^t \sqrt{3}\right) = 7\sqrt{3} e^t
\]

For the \(\mathbf{k}\) component:
\[
\frac{d}{dt}\left(7e^t \cos t\right) = 7e^t \cos t - 7e^t \sin t = 7e^t (\cos t - \sin t)
\]

Thus,
\[
\mathbf{v}(t) = \left(7e^t (\sin t + \cos t)\right) \mathbf{i} + \left(7\sqrt{3} e^t\right) \mathbf{j} + \left(7e^t (\cos t - \sin t)\right) \mathbf{k}
\]

### Step 2: Compute the acceleration vector \(\mathbf{a}(t)\)
The acceleration vector \(\mathbf{a}(t)\) is the derivative of the velocity vector \(\mathbf{v}(t)\) with respect to time \(t\):
\[
\mathbf{a}(t) = \frac{d\mathbf{v}(t)}{dt}
\]

Compute the derivative component-wise:
\[
\mathbf{a}(t) = \left(\frac{d}{dt}\left(7e^t (\sin t + \cos t)\right)\right) \mathbf{i} + \left(\frac{d}{dt}\left(7\sqrt{3} e^t\right)\right) \mathbf{j} + \left(\frac{d}{dt}\left(7e^t (\cos t - \sin t)\right)\right) \mathbf{k}
\]

For the \(\mathbf{i}\) component:
\[
\frac{d}{dt}\left(7e^t (\sin t + \cos t)\right) = 7e^t (\sin t + \cos t) + 7e^t (\cos t - \sin t) = 7e^t (2\cos t)
\]

For the \(\mathbf{j}\) component:
\[
\frac{d}{dt}\left(7\sqrt{3} e^t\right) = 7\sqrt{3} e^t
\]

For the \(\mathbf{k}\) component:
\[
\frac{d}{dt}\left(7e^t (\cos t - \sin t)\right) = 7e^t (\cos t - \sin t) - 7e^t (\sin t + \cos t) = -7e^t (2\sin t)
\]

Thus,
\[
\mathbf{a}(t) = \left(14e^t \cos t\right) \mathbf{i} + \left(7\sqrt{3} e^t\right) \mathbf{j} - \left(14e^t \sin t\right) \mathbf{k}
\]

### Step 3: Evaluate \(\mathbf{a}(0)\)
Substitute \(t = 0\) into \(\mathbf{a}(t)\):
\[
\mathbf{a}(0) = \left(14e^0 \cos 0\right) \mathbf{i} + \left(7\sqrt{3} e^0\right) \mathbf{j} - \left(14e^0 \sin 0\right) \mathbf{k}
\]
\[
\mathbf{a}(0) = \left(14 \times 1 \times 1\right) \mathbf{i} + \left(7\sqrt{3} \times 1\right) \mathbf{j} - \left(14 \times 1 \times 0\right) \mathbf{k}
\]
\[
\mathbf{a}(0) = 14 \mathbf{i} + 7\sqrt{3} \mathbf{j} + 0 \mathbf{k}
\]

### Final Answer
The acceleration vector at \(t = 0\) can be written as:
\[
\mathbf{a}(0) = 14 \mathbf{T} + 7\sqrt{3} \mathbf{N}
\]

Would you like more details or have any other questions? 

Here are five related questions to explore further:
1. How do you compute the magnitude of the velocity vector?
2. What is the significance of the tangent and normal vectors in this context?
3. How do you calculate the curvature of a curve given its parametric equations?
4. How does the Frenet-Serret frame relate to the acceleration components?
5. Can you express \(\mathbf{r}(t)\) in a different coordinate system?

**Tip:** When dealing with parametric equations, always compute derivatives carefully to avoid errors in the final result.

Learn how to compute the acceleration vector \(\mathbf{a}(0)\) at \(t = 0\) without directly finding the tangent \(\mathbf{T}\) and normal \(\mathbf{N}\) vectors. Detailed step-by-step solution using vector calculus and parametric equations.

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