A dot product measures alignment and gives an orthogonal projection.
Subtract the proposed projection from u and dot the result with v: u·v−[(u·v)/(v·v)](v·v)=0. Thus the residual is perpendicular. The identity u·v=||u||||v||cosθ also gives an angle when both vectors are nonzero. Here u=(1,2,3).
A zero dot product means perpendicular directions only when both vectors are nonzero.