(v x p) . v = 0

vv = GenericVector[v] ; pv = GenericVector[p] ; vv // Nice pv // Nice

( v ) x v y v z

( p ) x p y p z

tst = Cross[vv, pv] . vv ; tst // Nice

(p_y v_x - p_x v_y) v_z + v_y (p_x v_z - p_z v_x) + v_x (p_z v_y - p_y v_z)

Simplify[tst]

0

Created by Mathematica  (September 7, 2003)