Homework 6

Due Date:  Friday, October 17

1. A parametric curve \(\vec{x}(t)\) satisfies
   • \(\vec{x}(0) = (3,1,0)\),
   • \(\vec{x}\,'(0) = (2,1,2)\), and
   • \(\vec{x}\,''(0) = (12,9,6)\).
   1. Compute \(s\hspace{0.08333em}'(0)\) and \(s\hspace{0.08333em}''(0)\).
   2. Compute the Frenet frame \(\{\vec{T},\vec{P},\vec{B}\}\) at \(t=0\).
   3. Compute the curvature of the curve at \(t=0\).
   4. Find parametric equations for the osculating circle to the curve at \(t=0\).
   5. Find a Cartesian equation for the osculating plane to the curve at \(t=0\).
   6. Given that \(\vec{x}\,'''(0) = (11,-4,3)\), compute the torsion of the curve at \(t=0\).
2. Let \(\vec{x}(t)\) be a unit-speed curve satisfying
   • \(\vec{x}\,'(0) = (2/7,-3/7,6/7)\),
   • \(\vec{P}\hspace{0.083333em}'(0) = (4,0,-6)\), and
   • \(\tau(0)=-6\).
   1. Compute the curvature \(\kappa(0)\).
   2. Compute the binormal vector \(\vec{B}(0)\).
   3. Compute the principal normal vector \(\vec{P}(0)\).
   4. Given that \(\tau\hspace{0.08333em}'(0) = 14\), compute \(\vec{B}\hspace{0.083333em}''(0)\).