We got two vectors ##\mathbf{v_1}## and ##\mathbf{v_2}##, their sum is, geometrically, :
Now, let us rotate the triangle by angle ##\phi## (is this type of things allowed in mathematics?)
OC got rotated by angle ##\phi##, therefore ##OC' = T ( \mathbf{v_1} + \mathbf{v_2})##, and similarly...