A geometry problem by Hosam Hajjir

Collimated (parallel) rays of light fall on a circular ring of radius $5$ , centered at $(-5, -5, 10)$ , and lying in a plane whose normal is $(0, 1, 0)$ creating a shadow on the plane passing through $(5, 12, -7)$ , and whose normal is $(-3, -2, 5)$ .

The direction vector of the light rays is $(1, -2, -0.5)$ .

Find the unit vector along the major axis of the elliptical shadow of the ring (which has a non-negative $z$ -coordinate)