Solution: (1) According to the topic, we know that the coordinates of point A are (a, 0), so the coordinates of vectors OA and OB are (a, 0) and (1, 1). Then vector AB = vector o B- vector OA, and its coordinates are (1-a, 1). According to the meaning of the question, get

Vector OA? The vector ab = a (1-a)+0x1= a (1-a) =-2.

Solution: a =-1 (because the topic is A >；; 0, so give up), or a = 2.

Substituting a =2 and the point (1, 1) into the elliptic equation, we can get: b? = 4/3 。

So the equation of ellipse e is: x? /4 + y？ /(4/3) = 1 。

(2) According to the topic, it is easy to know that the equation of the straight line L is y = (1/3) (x-2);

The equation of straight line OB is y = X.

Two linear equations are simultaneous, and the coordinates of the intersection point are (-1,-1), which is the symmetrical point of point B about the origin. According to the symmetry of the ellipse about the origin, the coordinates of the intersection (-1,-1) must be on the ellipse e, that is, it is the point C. So the points B, O and C*** lines.