Although we have derived the entries for in terms of the angle θ, it is not necessary that we do so. An alternative approach, and one that scales nicely to the three-dimensional case, is to build the rotation matrix by projecting the axes of frame o1x1y1 onto the coordinate axes of frame o0x0y0. Recalling that the dot product of two unit vectors gives the projection of one onto the other, we obtain
which can be combined to obtain the rotation matrix
Thus, the columns of specify the direction cosines of the coordinate axes of o1x1y1 relative to the coordinate axes of o0x0y0. For example, the first column (x1 · x0, x1 · y0) of
specifies the direction of x1 relative to the frame o0x0y0. Note that the right-hand sides of these equations are defined in terms of geometric entities, and not in terms of their coordinates. Examining Figure 2.2 it can be seen that this method of defining the rotation matrix by projection gives the same result as we obtained in Equation (2.1).
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 and it is easily shown that the column vectors of
are of unit length and mutually orthogonal (Problem 2–4). Thus
is an orthogonal matrix. It also follows from the above that (Problem 2–5)
. If we restrict ourselves to right-handed coordinate frames, as defined in Appendix B, then
(Problem 2–5).
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 following properties hold
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 is rotated through an angle θ about the z0-axis, and we wish to find the resulting transformation matrix . By convention, the right hand rule (see Appendix B) defines the positive sense for the angle θ to be such that rotation by θ about the z-axis would advance a right-hand threaded screw along the positive z-axis.
From Figure 2.3 we see that
and
while all other dot products are zero. Thus, the rotation matrix has a particularly simple form in this case, namely
(2.3)