College Algebra Exam Review 214

College Algebra Exam Review 214 - A reection in R 3 through...

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224 4. SYMMETRIES OF POLYHEDRA Show that f 2 4 0 0 ˙ p 5=2 3 5 g [ A [ B is the set of vertices of an icosahedron. 4.2.3. Each vertex of the icosahedron lies on a 5–fold axis, each midpoint of an edge on a 2–fold axis, and each centroid of a face on a 3–fold axis. Using the data of the previous exercise and the method of Exercises 4.1.1 , 4.1.2 , and 4.1.3 , you can compute the matrices for rotations of the icosahe- dron. (I have only done this numerically and I don’t know if the matrices have a nice closed form.) 4.3. What about Reflections? When you thought about the nature of symmetry when you first began reading this text, you might have focused especially on reflection symme- try. (People are particularly attuned to reflection symmetry since human faces and bodies are important to us.)
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Unformatted text preview: A reection in R 3 through a plane P is the transformation that leaves the points of P xed and sends a point x 62 P to the point on the line through x and perpendicular to P , which is equidistant from P with x and on the opposite side of P . Figure 4.3.1. A reection. For a plane P through the origin in R 3 , the reection through P is given by the following formula. Let be a unit vector perpendicular to P . For any x 2 R 3 , the reection j of x through P is given by j . x / D x 2 h x ; i , where h ; i denotes the inner product in R 3 . In the Exercises, you are asked to verify this formula and to compute the matrix of a reection, with respect to the standard basis of R 3 . You are...
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