By free, I mean not connected to the other vector. In fact any number of vectors may be added graphically by creating a chain of the vectors, in any order, head to tail. Remember that the absolute position of a vector is meaningless, and that its length and angle carry its meaning. This is equivalent to adding to for that vector. The graphical method is useful for developing intuition about vector addition, but we will use a mathematical method in most calculations.
Vector addition is the scalar addition of the Cartesian components of the vectors.
There is no convenient way to add vectors in terms of their polar coordinates, so you will always want to convert when you add or subtract. Multiplication of a vector by a scalar affects only the magnitude of the vector. You simply multiply the magnitude by the scalar to find the new magnitude, and leave the angle alone. In Cartesian coordinates, you simply multiply both components by the scalar. It should make sense that the direction of a vector cannot be changed by multiplication by a scalar, since scalars have no direction.