/* --------------------------------- */
/* save as x_v3d.h                   */
/* --------------------------------- */
/* --------------------------------- */
double norm3d(
v3d u
)
{
       u.i *= u.i;
       u.j *= u.j;
       u.k *= u.k;

 return(sqrt(u.i+u.j+u.k));
}
/* --------------------------------- */
v3d normalize3d(
v3d u
)
{
v3d v;
double   d;

         d = norm3d(u);

       v.i = u.i/d;
       v.j = u.j/d;
       v.k = u.k/d;

 return(v);
}
/* --------------------------------- */
double dotproduct3d(
v3d u,
v3d v
)
{
 return( (u.i*v.i + u.j*v.j + u.k*v.k) );
}
/* --------------------------------- */
v3d mns_u3d(
v3d u
)
{
v3d v;

       v.i = -u.i;
       v.j = -u.j;
       v.k = -u.k;

 return(v);
}
/* --------------------------------- */