We begin our study of the representation of the configuration of a rigid body by focusing on orientation only. The approach to representing the full configuration of a rigid body is analogous.

Consider two frames, a space frame {s} and a body frame {b}. They are shown at different locations, but we are focusing on their orientations. We can express the orientation of the frame {b} relative to {s} by writing the unit coordinate axes of frame {b} in the coordinates of frame {s}. In the coordinates of {s}, the x_b-axis is 0, 1, 0, the y_b-axis is -1, 0, 0, and the z_b-axis is 0, 0, 1. We can write these column vectors side by side to form the rotation matrix R_sb. The second subscript, {b}, indicates the frame whose orientation is being represented, and the first subscript, {s}, is the frame of reference.

Sometimes the two subscripts are implicit and we leave them out, writing the rotation matrix simply as R.

As we learned in Chapter 2, the space of orientations of a rigid body is only 3 dimensional, but we have 9 numbers in a rotation matrix. That means the 9 entries of the matrix must be subject to 6 constraints. Three of those constraints are that the column vectors are all unit vectors, and the other 3 are that the dot product of any two of the column vectors is zero. In other words, the 3 vectors are orthogonal to each other. These 6 constraints can be written compactly as R transpose times R is equal to the 3 by 3 identity matrix I. These constraints ensure that the determinant of R is either 1, corresponding to right-handed frames, or -1, corresponding to left-handed frames. We only use right-handed frames, so the determinant of R must be 1.

The set of all rotation matrices is called the special orthogonal group SO(3): the set of all 3x3 real matrices R such that R transpose R is equal to the identity matrix and the determinant of R is equal to 1.

Rotation matrices satisfy the following properties: The inverse of R is equal to its transpose, which is also a rotation matrix. The matrix product of two rotation matrices is also a rotation matrix. Matrix multiplication is associative, but in general it is not commutative. Finally, for any 3-vector x, R times x has the same length as x. As we will see later, this means that rotating a vector does not change its length.

In the next video, we will study 3 common uses of rotation matrices.