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Let's go through the problem in parts as requested. 

### Problem Breakdown and Solution Guide

**Q2 Overview**
- The problem involves calculating the binormal vector **B** and torsion **τ** for given curves. It consists of four parts, which each ask for different aspects of these curves and their properties.
  
#### Definitions and Concepts:
- **Binormal Vector (B)**: For a curve \( r(t) \), the binormal vector \( B \) is given by \( B = T \times N \), where \( T \) is the unit tangent vector, and \( N \) is the unit normal vector.
  
- **Torsion (τ)**: Measures how much the curve twists out of the plane of curvature. It is given by:
  \[
  \tau = -\frac{1}{|v|} \frac{dB}{dt} \cdot N
  \]
  where \( v = \frac{dr}{dt} \) is the velocity vector.

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### Solutions

#### Part (a)
**Objective**: Show that the helix \( q(t) = \langle a \cos(t), a \sin(t), b t \rangle \) has constant torsion.

1. **Calculate \( T \) and \( N \)**:
   - Compute \( v = \frac{dq}{dt} \).
   - Find \( |v| \) and determine \( T = \frac{v}{|v|} \).
   - Differentiate \( T \) with respect to \( t \) to get \( N \), and then compute \( B = T \times N \).

2. **Compute Torsion** \( τ \):
   - Using the formula for torsion above, check if \( τ \) is constant.

This part shows that a helix has constant non-zero curvature and torsion, which is a unique property.

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#### Part (b)
**Objective**: Sketch the curve \( C \) given by \( r(t) = \langle 9 \cos(t), 9 \sin(t), e^{-t} \rangle \).

- This curve spirals in the \( xy \)-plane with radius 9 while moving down in the \( z \)-direction because \( z = e^{-t} \) decays to zero as \( t \to \infty \).
- Roughly sketch a spiral in the \( xy \)-plane that shrinks toward \( z = 0 \) as \( t \) increases.

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#### Part (c)
**Objective**: Attempt to compute the torsion \( τ \) for \( C \) and recognize that it is challenging.

- Calculating torsion for this curve will be complex due to the exponential term in \( z \)-coordinate.
- Recognize the challenge of directly computing \( τ \) here, as hinted in the problem instructions.

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#### Part (d)
**Objective**: Describe the behavior of vectors \( T \), \( N \), \( B \), curvature \( κ \), and torsion \( τ \) as \( t \to \infty \) for \( C \).

1. **As \( t \to \infty \)**:
   - The \( z \)-component \( e^{-t} \) approaches zero, so the curve flattens in the \( xy \)-plane.
   - \( T \) will point tangentially along the spiral in the \( xy \)-plane.
   - \( N \) will likely stabilize, remaining in the plane of the curve.
   - \( B \) becomes perpendicular to both \( T \) and \( N \), eventually aligning more with the \( z \)-axis as the curve flattens.
   - **Curvature \( κ \)**: As the curve becomes planar, curvature may stabilize.
   - **Torsion \( τ \)**: Torsion approaches zero as the curve flattens.

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Would you like further detail on any of these steps or need help with the calculations?

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### Follow-Up Questions

1. What would be the steps to find the unit tangent vector \( T \) for \( q(t) \)?
2. How do we compute the normal vector \( N \) and the binormal vector \( B \) for a helical curve?
3. What would be the geometric interpretation of constant curvature and torsion?
4. How would we evaluate torsion for a complex curve like \( C \) in a practical scenario?
5. Why does torsion tend to zero as a curve flattens in a plane?

### Tip
For curves with exponential or trigonometric terms, breaking down the components (like computing derivatives in stages) can simplify finding vectors like \( T \), \( N \), and \( B \).

The binormal vector to a curve r(t) is B = T x N. We can also calculate B using the formula B = (v x a) / |v x a|. The problem has four parts: (a) Show that the helix q(t) = <a cos(t), a sin(t), b t> has constant torsion. (b) Sketch the curve C given by r(t) = <9 cos(t), 9 sin(t), e^(-t)>. (c) Attempt to compute the torsion τ of C, recognizing its difficulty. (d) Describe what happens to T, N, B, κ, and τ as t goes to infinity.

Explore the torsion and binormal vector calculations for parametric curves, including the unique properties of a helical curve with constant torsion. This advanced calculus problem involves binormal vector formula derivation, torsion computation, and vector analysis as t approaches infinity.

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