  Let R be a regular semi-local domain containing a field such that all the
residue fields are infinite. Let K be the fraction field of R. Let q be a
quadratic space over R on a free rank n R-module P such that the projective
quadric q=0 is smooth over R. It is proved that if the quadratic space q is
isotropic over K, then there is a unimodular vector v in the free rank n
R-module P such that q(v)=0.
  If characteristic of R is 2, then in the case of even n our assumption on q
is equivalent to the one that q is a non-singular space in the sense of
\cite{Kn} and in the case of odd n > 2 our assumption on q is equivalent to the
one that q is a semi-regular in the sense of \cite{Kn}.

Algebraic Geometry