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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 righthanded rotation through the angle 2 =3 about
1 ...
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 Fall '08
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 Algebra

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