wish i had a pen and paper.
still , see if this has anything useful(read it in penrose's book)
imagine a flat solid body in our familiar xy plane. first rotate about z axis through an angle alpha. so x and y axes also get shifted but remain in the same plane.(make these lines bold and rub previous ones)
now about new y axis (bold) rotate it by angle beta. now the plane gets tilted.(again rub previous ones and keep the axis about which rotation is done ,fixed)
do the same with the new bold z axis.(rub previous axes) by angle gamma, we get the final co ordinate system.represent each rotation by a matrix, product of the 3 matrices gives us the final direction cosines)
i found a treatment in a book called arfken and weber. if u want a complete prescription, that book is OK(but to me its a dangerous book half the time i dont understand what is done )
