Mathematical Methods for Physicists: A concise introduction - Site Map

ROTATION MATRICES
Eq. (3.45) gives six relations among the ij , and is known as the orthogonal
condition.
If the primed coordinates system is generated by a rotation about the x 3 -axis
through an angle as shown in Fig. 3.2. Then from Example 3.5, we have
Thus
x 0 1 ˆ x 1 cos ‡ x 2 sin ; x 0 2 ˆx 1 sin ‡ x 2 cos ; x 0 3 ˆ x 3 : …3:46†
11 ˆ cos ; 12 ˆ sin ; 13 ˆ 0;
21 ˆsin ; 22 ˆ cos ; 23 ˆ 0;
31 ˆ 0; 32 ˆ 0; 33 ˆ 1:
We can also obtain these elements from Eq. (3.42a). It is obvious that only three
of them are independent, and it is easy to check that they satisfy the condition
given in Eq. (3.45). Now the rotation matrix takes the simple form
0
1
cos sin 0
B
C
~…† ˆ@
sin cos 0 A
…3:47†
0 0 1
and its transpose is
0
1
cos sin 0
~ T B
…† ˆ@
sin cos
C
0 A:
0 0 1
Now take the product
0
cos sin
10
0 cos sin
1
0
0
1 0
1
0
~ T …† …† ~ B
ˆ@
sin cos
CB
0 A@
sin cos
C B
0 A ˆ @ 0 1
C
0A ˆ ~I;
0 0 1 0 0 1 0 0 1
which shows that the rotation matrix is an orthogonal matrix. In fact, rotation
matrices are orthogonal matrices, not limited to …† ~ of Eq. (3.47). The proof of
this is easy. Since coordinate transformations are reversible by interchanging old
and new indices, we must have
ij
ˆ ^e old
~ 1
i
^e new
j
ˆ ^e new
j
^e old
i ˆ ji ˆ ~ T
Hence rotation matrices are orthogonal matrices. It is obvious that the inverse of
an orthogonal matrix is equal to its transpose.
A rotation matrix such as given in Eq. (3.47) is a continuous function of its
argument . So its determinant is also a continuous function of and, in fact, it is
equal to 1 for any . There are matrices of coordinate changes with a determinant
of 1. These correspond to inversion of the coordinate axes about the origin and
119
ij :