  We show that the superposition principle applies to coupled nonlinear
Schr\"odinger equations with cubic nonlinearity where exact solutions may be
obtained as a linear combination of other exact solutions. This is possible due
to the cancellation of cross terms in the nonlinear coupling. First, we show
that a {\it composite} solution which is a linear combination of the two
components of a {\it seed} solution is another solution to the same coupled
nonlinear Schr\"odinger equation. Then, we show that a linear combination of
two composite solutions is also a solution to the same equation. With emphasis
on the case of Manakov system of two-coupled nonlinear Schr\"odinger equations,
the superposition is shown to be equivalent to a rotation operator in a
two-dimensional function space with components of the seed solution being its
coordinates. Repeated application of the rotation operator, starting with a
specific seed solution, generates a series of composite solutions which may be
represented by a generalized solution that defines a family of composite
solutions. Applying the rotation operator to almost all known exact seed
solutions of the Manakov system, we obtain for each seed solution the
corresponding family of composite solutions. Composite solutions turn out, in
general, to possess interesting features that do not exist in the seed
solution. Using symmetry reductions, we show that the method applies also to
systems of $N$-coupled nonlinear Schr\"odinger equations. Specific examples for
the three coupled-nonlinear Schr\"odinger equation are given.
