Homework 8

Due Date: Friday, October 31

1. Let \(H\) be the hyperplane \(x_1 + x_2 + x_3 + x_4 = 4\) in \(\mathbb{R}^4\). Find a parametrization for the (two-dimensional) unit sphere on \(H\) centered at the point \((1,1,1,1)\).
2. The paraboloid \(x_3 = x_1^2 + x_2^2\) on the \(x_1x_2x_3\)-hyperplane in \(\mathbb{R}^4\) is reflected across the hyperplane \(x_1+x_2+x_3+x_4 = 1\). Find parametric equations for the resulting surface.
3. 1. Find formulas (in terms of \(u\) and \(v\)) for two perpendicular unit vectors that are both perpendicular to the vector \((\cos u \sin v,\sin u \sin v,\cos v)\).
      Hint: Use tangent vectors to the unit sphere.
   2. Let \(M\) be the \(3\)-manifold in \(\mathbb{R}^6\) consisting of all points \((x_1,x_2,x_3,x_4,x_5,x_6)\) for which \((x_1,x_2,x_3)\) and \((x_4,x_5,x_6)\) are perpendicular unit vectors in \(\mathbb{R}^3\). Find a parametrization of \(M\).
4. The equation \[ \left(\sqrt{{x_1}^2 + {x_2} ^2 + {x_3}^2} \,-\, 2\right)^2 \,+\, {x_4}^2 \,+\, {x_5}^2 \;=\; 1 \] defines a “sphere of spheres” in \(\mathbb{R}^5\). Find a parametrization of this \(4\)-manifold.
   Hint: Consider the torus \(\displaystyle\left(\sqrt{x^2+y^2} \hspace{0.08333em}-\hspace{0.08333em} 2\right)^2 + z^2 \,=\, 1\) in \(\mathbb{R}^3\).