Math 312 Bard College

Homework 1

Due Date: Friday, September 15

Instructions: Feel free to work together with other students in the class, though you must turn in your own copy of the solutions, and you must acknowledge anyone that you worked with.

  1. The following figure shows a polyhedron in \(\mathbb{R}^3\) with six vertices, nine edges, and five faces.
    1. What is the volume of this polyhedron? Explain.
    2. Find the point on the surface of this polyhedron that lies closest to the point (−1,3,4). Explain your answer.
  2. In the following triangle, \(P\) is the midpoint of \(\overline{BC}\), and \(|A-Q| = 2|C-Q|\). Express the point \(R\) as a linear combination of the points \(A\), \(B\), and \(C\). Explain your reasoning.
  3. Let \(P\) be the plane in \(\mathbb{R}^4\) that goes through the points (0,0,0,0), (1,0,−1,1), and (0,1,1,2).
    1. Find the point on \(P\) that lies closest to the point (−5,4,0,0). Justify your answer.
    2. Find the equation for the hyperplane in \(\mathbb{R}^4\) that contains the plane \(P\) as well as the point (1,1,1,1).