that we will discuss below.
In the two-dimensional case, it is straightforward to compute the entries of this matrix. As illustrated in Figure 2.2,
which gives
(2.1)
Note that we have continued to use the notational convention of allowing the superscript to denote the reference frame. Thus,
Although we have derived the entries for
which can be combined to obtain the rotation matrix
Thus, the columns of
If we desired instead to describe the orientation of frame o0x0y0 with respect to the frame o1x1y1 (that is, if we desired to use the frame o1x1y1 as the reference frame), we would construct a rotation matrix of the form
Since the dot product is commutative, (that is, xi · yj = yj · xi), we see that
In a geometric sense, the orientation of o0x0y0 with respect to the frame o1x1y1 is the inverse of the orientation of o1x1y1 with respect to the frame o0x0y0. Algebraically, using the fact that coordinate axes are mutually orthogonal, it can readily be seen that
The above relationship implies that
More generally, these properties extend to higher dimensions, which can be formalized as the so-called special orthogonal group of order n.
Definition 2.1.
The special orthogonal group of order n, denoted SO(n), is the set of n × n real-valued matrices
(2.2)
Thus, for any
The columns (and therefore the rows) of are mutually orthogonal
Each column (and therefore each row) of is a unit vector
The special case, SO(2), respectively, SO(3), is called the rotation group of order 2, respectively 3.
To provide further geometric intuition for the notion of the inverse of a rotation matrix, note that in the two-dimensional case, the inverse of the rotation matrix corresponding to a rotation by angle θ can also be easily computed simply by constructing the rotation matrix for a rotation by the angle − θ:
2.2.2 Rotations in Three Dimensions
The projection technique described above scales nicely to the three-dimensional case. In three dimensions, each axis of the frame o1x1y1z1 is projected onto coordinate frame o0x0y0z0. The resulting rotation matrix R ∈ SO(3) is given by
As was the case for rotation matrices in two dimensions, matrices in this form are orthogonal, with determinant equal to 1 and therefore elements of SO(3).
Example 2.1.
Suppose the frame o1x1y1z1
