Introduce unit vectors ı ı ^ ^ , ȷ ȷ ^ ^ , k ^ ^ , now representing standard unit basis vectors in the general rotating frame. As they rotate they will remain normalized and perpendicular to each other. If they rotate at the speed of Ω Ω ( t ) about an axis along the rotation vector Ω Ω ( t ) then each unit vector u ^ ^ of the rotating coordinate system (such as ı ı ^ ^ , ȷ ȷ ^ ^ , or k ^ ^ ) abides by the following equation: d d t u ^ ^ = Ω Ω × × u ^ ^ . So if R ( t ) denotes the transformation taking basis vectors of the inertial- to the rotating frame, with matrix columns equal to the basis vectors of the rotating frame, then the cross product multiplication by the rotation vector is given by Ω Ω × × = R ′ ( t ) ⋅ ⋅ R ( t ) T .