A matrix that rotates a vector
The standard way to represent orientation in 3D robotics is the rotation matrix, a 3x3 grid of numbers. Conceptually, a rotation matrix is a rule: feed it a vector and it hands back that same vector rotated into a new direction, with its length preserved. Stack up the directions a frame's own axes point in, expressed relative to some reference frame, and you get exactly that 3x3 matrix.
This makes rotation matrices very direct to work with. Rotating a vector is a single, well-defined operation, and combining two rotations, first rotate this way, then rotate that way, is done by combining their matrices. That directness is a big part of why rotation matrices remain the standard building block in robotics libraries and formulas, even though, as later sections show, they aren't the most compact way to store an orientation.