Rotation does not alter the length of the vector. Consequently, the transformation matrices are orthogonal, real and also unitary matrices, i.e. \(R^{\dagger } R =1\) . These formulas represent active rotations, where the coordinate system is fixed and the vector is rotated. In a passive rotation, the vector is fixed and coordinate system is rotated. The corresponding matrices for passive rotations can be obtained by \(\theta \rightarrow -\theta \) substitutions.