https://file.aiquickdraw.com/mathbot/file/1730413868101okq7k2v8.jpg

Let's break down each part of the question for clarity:

1. **Torsion and the Binormal Vector**: 
   - The problem defines the **binormal vector** \(\mathbf{B} = \mathbf{T} \times \mathbf{N}\) for a curve \(\mathbf{r}(t)\).
   - There’s also an alternative formula for \(\mathbf{B}\):  
     \[
     \mathbf{B} = \frac{\mathbf{v} \times \mathbf{a}}{|\mathbf{v} \times \mathbf{a}|}
     \]
     where \(\mathbf{v}\) is the velocity (first derivative of \(\mathbf{r}(t)\)) and \(\mathbf{a}\) is the acceleration (second derivative of \(\mathbf{r}(t)\)).
   - Torsion, denoted \(\tau\), measures how much the curve "twists" out of the plane of curvature and is defined as:
     \[
     \tau = -\frac{1}{|\mathbf{v}|} \frac{d\mathbf{B}}{dt} \cdot \mathbf{N}
     \]

### Part (a)
You are asked to **prove that the helix \(\mathbf{q}(t) = \langle a \cos(t), a \sin(t), bt \rangle\)** has constant torsion. 

For this:
   - Compute the velocity vector \(\mathbf{v} = \frac{d\mathbf{q}}{dt}\) and acceleration vector \(\mathbf{a} = \frac{d^2\mathbf{q}}{dt^2}\).
   - Use these vectors to compute the torsion \(\tau\) and verify that it is constant.

### Part (b)
This part introduces a new curve, **\(\mathbf{r}(t) = \langle 9 \cos(t), 9 \sin(t), e^{-t} \rangle\)**, and asks for a **rough sketch** of it. 
   - Notice that the first two components \((9 \cos(t), 9 \sin(t))\) define a circular motion in the \(xy\)-plane with radius 9.
   - The \(z\)-component, \(e^{-t}\), decreases as \(t\) increases, suggesting a spiral that moves downward toward the \(xy\)-plane.

### Part (c)
You are instructed to compute the torsion \(\tau\) of the curve \(C\) from part (b). However, the problem hints that this is difficult and suggests you acknowledge the challenge instead of performing full calculations.

### Part (d)
This part requires reasoning about the behavior of the **tangent** (\(\mathbf{T}\)), **normal** (\(\mathbf{N}\)), **binormal** (\(\mathbf{B}\)), **curvature** (\(k\)), and **torsion** (\(\tau\)) vectors as \(t \to \infty\) for the curve \(\mathbf{r}(t)\) from part (b).

- Intuitively, as \(t \to \infty\), \(e^{-t} \to 0\), so the curve flattens towards the \(xy\)-plane, impacting the torsion and curvature values. You would argue about the limits of these values based on the curve's shape and behavior.

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### Summary of Steps for Solution:

1. **Part (a)**: Calculate derivatives of \(\mathbf{q}(t)\), determine torsion.
2. **Part (b)**: Sketch the path based on circular motion and decaying \(z\)-component.
3. **Part (c)**: Attempt the torsion calculation; note the difficulty.
4. **Part (d)**: Discuss vector behaviors as \(t \to \infty\), focusing on geometry.

Would you like a detailed solution for each part?

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### Follow-Up Questions
1. What is the significance of a constant torsion in a curve?
2. How do we interpret the binormal vector \(\mathbf{B}\) in the context of the curve's geometry?
3. Can we generalize the behavior of a curve as \(t \to \infty\) for different parameterizations?
4. How does curvature change for different types of curves (e.g., circular vs. helical)?
5. Why might calculating torsion be challenging for certain parameterizations?

### Tip
When dealing with curves in space, visualizing the trajectory by plotting or sketching can clarify how components like torsion and curvature behave.

Q2 (10 points):
The binormal vector to a curve r(t) is B = T × N. We can also calculate B using the (usually more convenient) formula B = (v × a) / |v × a|.

(a) Show that the helix q(t) = ⟨a cos(t), a sin(t), bt⟩ has constant torsion.

(b) Let C be the curve parametrized by r(t) = ⟨9 cos(t), 9 sin(t), e^(-t)⟩. Draw a rough sketch of C.

(c) Attempt to compute the torsion, τ, of C. Write "wow, this is very difficult!" on your submission for this part to receive full credit.

(d) For the curve C, describe what happens to the vectors and values as t goes to infinity: T, N, B, k, and τ. Use geometric reasoning for this problem.

Explore the torsion and curvature properties of a helical curve with constant torsion, and an exponential decay curve, with step-by-step reasoning and explanations. Ideal for students studying vector calculus at the undergraduate level.

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