College Algebra Exam Review 204

# College Algebra Exam Review 204 - 214 4 SYMMETRIES OF...

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Unformatted text preview: 214 4. SYMMETRIES OF POLYHEDRA Figure 4.1.4. Right–hand rule. O If v is the ﬁrst standard coordinate vector 23 1 O e1 D 405; 0 then the rotation matrix is 2 1 RÂ D 40 0 0 cos Â sin Â 3 0 sin Â 5 : cos Â O To compute the matrix for the rotation about the vector v, ﬁrst we need O O O OO O to ﬁnd additional vectors v2 and v3 such that fv1 D v; v2 ; v3 g form a right–handed orthonormal basis of R3 ; that is, the three vectors are of unit length, mutually orthogonal, and the determinant of the matrix V D OOO O O Œv1 ; v2 ; v3  with columns vi is 1, or equivalently, v3 is the vector crossO O product v1 v2 . V is the matrix that rotates the standard right–handed OOO OOO orthonormal basis fe1 ; e2 ; e3 g onto the basis fv1 ; v2 ; v3 g. The inverse of t , because the matrix entries of V t V are the V is the transposed matrix V OO inner products hvi ; vj i D ıij . The matrix we are looking for is VRÂ V t , O because the matrix ﬁrst rotates the orthonormal basisfvi g onto the standard O O orthonormal basisfei g, then rotates through an angle Â about e1 , and then O O rotates the standard basis fei g back to the basis fvi g. Consider the points 23 23 23 23 1 1 1 1 415 ; 4 15 ; 4 15 ; and 4 15 : 1 1 1 1 They are equidistant from each other, and the sum of the four is 0, so the four points are the vertices of a tetrahedron whose center of mass is at the origin. One 3 2 3–fold axis of the tetrahedron passes through the origin and the 1 point 415: Thus the right-handed rotation through the angle 2 =3 about 1 ...
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