sometimes, i will confuse on changing frame, especially between rotating frame and lab frame.
the notation plays an important role. for a rotating vector.
where is the static frame vector. is the rotation operator, with right -hand rotation is positive around direction . for clearance, when we think on a rotating, is always positive.
however, if we regard the vector does not rotate but is the frame rotating, which is equivalent as rotating backward ( left-hand or negative ). the expression is the same but with different notation.
is the rotating frame vector, is the rate of rotating of the rotating frame respect to the static frame. for the rotating frame is rotating forward ( positive)
if both the vector is rotating an the rotating frame is rotating, in same direction, with different rate.
and combine the rotation operator.
which make perfect sense.
the principle can apply to any frame transformation.
__________________ Examples _____________________
a rotation on positive direction for the vector.
if we add the frame reference on as a subscript, things will be much clear. since the same vector can have different coordinate expression. so, we better mark the coordinate with reference system.