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.
