College Algebra Exam Review 215

College Algebra Exam Review 215 - 4.3. WHAT ABOUT...

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Unformatted text preview: 4.3. WHAT ABOUT REFLECTIONS? 225 also asked to find a formula for the reflection through a plane that does not pass through the origin. A reflection that sends a geometric figure onto itself is a type of symmetry of the figure. It is not an actual motion that you could perform on a physical model of the figure, but it is an ideal motion. Let’s see how we can bring reflection symmetry into our account of the symmetries of some simple geometric figures. Consider a thickened version of our rectangular card: a rectangular brick. Place the brick with its faces parallel to the coordinate planes and with its centroid at the origin of coordinates. See Figure 4.3.2. r 3 r 1 r 2 Figure 4.3.2. Rotations and reflections of a brick. The rotational symmetries of the brick are the same as those of the rectangular card. There are four rotational symmetries: the nonmotion e , and the rotations r1 ; r2 , and r3 through an angle of about the x –, y –, and z –axes. The same matrices E , R1 , R2 , and R3 listed in Section 1.5 implement these rotations. In addition, the reflections in each of the coordinate planes are symmetries; write ji for jei , the reflection in the plane orthogonal to the standard O O unit vector ei . See Figure 4.3.2. The symmetry ji is implemented by the diagonal matrix Ji with 1 in the i t h diagonal position and 1’s in the other diagonal positions. For example, 2 3 1 00 1 05 : J2 D 40 0 01 There is one more symmetry that must be considered along with these, which is neither a reflection nor a rotation but is a product of a rotation ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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