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Recall that the matrix in Equation (2.9) represents a rotational transformation between the frames o₀x₀y₀z₀ and o₁x₁y₁z₁. Suppose we now add a third coordinate frame o₂x₂y₂z₂ related to the frames o₀x₀y₀z₀ and o₁x₁y₁z₁ by rotational transformations. A given point p can then be represented by coordinates specified with respect to any of these three frames: , , and The relationship among these representations of p is

(2.13)

(2.14)

(2.15)

where each is a rotation matrix. Substituting Equation (2.14) into Equation (2.13) gives

(2.16)

Note that and represent rotations relative to the frame o₀x₀y₀z₀ while represents a rotation relative to the frame o₁x₁y₁z₁. Comparing Equations (2.15) and (2.16) we can immediately infer

(2.17)

Equation (2.17) is the composition law for rotational transformations. It states that, in order to transform the coordinates of a point p from its representation in the frame o₂x₂y₂z₂ to its representation in the frame o₀x₀y₀z₀, we may first transform to its coordinates in the frame o₁x₁y₁z₁ using and then transform to using .

We may also interpret Equation (2.17) as follows. Suppose that initially all three of the coordinate frames coincide. We first rotate the frame o₁x₁y₁z₁ relative to o₀x₀y₀z₀ according to the transformation . Then, with the frames o₁x₁y₁z₁ and o₂x₂y₂z₂ coincident, we rotate o₂x₂y₂z₂ relative to o₁x₁y₁z₁ according to the transformation . The resulting frame, o₂x₂y₂z₂ has orientation with respect to o₀x₀y₀z₀ given by . We call the frame relative to which the rotation occurs the current frame.

Example 2.5.

Suppose a rotation matrix represents a rotation of angle ϕ about the current y-axis followed by a rotation of angle θ about the current z-axis as shown in Figure 2.8. Then the matrix is given by

(2.18)

Figure 2.8 Composition of rotations about current axes.

It is important to remember that the order in which a sequence of rotations is performed, and consequently the order in which the rotation matrices are multiplied together, is crucial. The reason is that rotation, unlike position, is not a vector quantity and so rotational transformations do not commute in general.

Example 2.6.

Suppose that the above rotations are performed in the reverse order, that is, first a rotation about the current z-axis followed by a rotation about the current y-axis. Then the resulting rotation matrix is given by

(2.19)

Comparing Equations (2.18) and (2.19) we see that .