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.

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.

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.

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.

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.

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

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 $B$.