ABSTRACT
In this chapter, it is shown that the norms and the inner products of vectors and the determinants of matrices are preserved by a 3D rotation. It is concluded that a 3D rotation is represented by a “rotation matrix” and has three degrees of freedom. The authors derive “Euler’s theorem,” which states that a 3D rotation is a rotation around some axis by some angle. They show expressions of 3D matrices that represent rotations around coordinate axes. Among rotations around axes, the most fundamental ones are those around the coordinate axes. In the following, the axis is regarded as oriented and the angle of rotation is regarded as positive or negative according to the right-hand rule. The 3D rotation plays a central role in analysis of dynamics of rigid bodies and angular momentum of elementary particles, and hence it has mostly been discussed by physicists.
