For the first eight or nine planets, what are the directions of the
eccentricity vectors, that is, the directions from the sun to the
perihelions of the planets.

I want to know to what extent the projections of the eccentricity
vectors on the Earth's ecliptic plane may be nearly all in the same
direction or may be seemingly randomly oriented.

The ideal answer is the angle of the projection of the eccentricity
vector of each planet on the Earth?s ecliptic plane, with respect to
the Earth?s eccentricity vector.

The ideal answer is in the form of a reference to a reliable web site.
 The next ideal is eight or nine angles, with a library or
encyclopedia reference.

Hello,

   The following link has orbital elements for the nine major planets:

http://ssd.jpl.nasa.gov/txt/p_elem_t1.txt

   These data are offered by the website as being of usable accuracy
between the years 1800 & 2050. The 5th column from the left has the
longitudes of the planets' perihelia. As you can see, the major axes
of the orbits are currently pointed over a large variety of
orientations.