The vector #vec w# is in the plane defined by #vec u# and #vec v#.
Unless #vec u = -vec v#, we have #vec w ne 0#.
Let #theta_u# and #theta_v# be the angles the vector #vec w# makes with #vec u# and #vec v#, respectively. Then we have
#vec w cdot vec u = |vec w| |vec u| cos theta_u #
#qquadqquad = (|vec u|vec v+|vec v| vec u)cdot vec u#
#qquad qquad = |vec u|(vec v cdot vec u)+|vec v| (vec u cdot vec u)#
#qquad qquad = |vec u|(vec v cdot vec u+|vec v| |vec u|) implies#
#|vec w| cos theta_u = vec v cdot vec u+|vec v| |vec u|#
By interchanging #vec u# and #vec v# we find
#|vec w| cos theta_v = vec u cdot vec v+|vec u| |vec v|#
and so
#|vec w| cos theta_u = |vec w| cos theta_v implies#
#color(red)(theta_u = theta_v)#
Thus #vec w# bisects the angle between #vec u# and #vec v# (except in the special case #vec u = -vec v# )
