Your work is sufficient to prove the result, yes.
given
vector U=(1,3,-2)
vector V=(-2,2,2)
vector W=(5,1,4)
demonstrate that vectors u,v, and w are all perpendicular to each other.
i took the dot product u dot v, v dot w, and u dot w and they all came to be 0.
is this complete after three dot products prove to be 0 demonstrating they are all perpendicular?