Thus the time derivative of these vectors, which rotate without changing magnitude, is d d t ı ı ^ ^ ( t ) = Ω Ω ( − − sin ⁡ ⁡ θ θ ( t ) , cos ⁡ ⁡ θ θ ( t ) ) = Ω Ω ȷ ȷ ^ ^ ; d d t ȷ ȷ ^ ^ ( t ) = Ω Ω ( − − cos ⁡ ⁡ θ θ ( t ) , − − sin ⁡ ⁡ θ θ ( t ) ) = − − Ω Ω ı ı ^ ^ , where Ω Ω ≡ ≡ d d t θ θ ( t ) . This result is the same as found using a vector cross product with the rotation vector Ω Ω pointed along the z-axis of rotation Ω Ω = ( 0 , 0 , Ω Ω ) , namely, d d t u ^ ^ = Ω Ω × × u ^ ^ , where u ^ ^ is either ı ı ^ ^ or ȷ ȷ ^ ^ .