Here:

http://www.euclideanspace.com/maths/geometry/elements/plane/lineOnPlane/

It is stated that:

"So the direction is:

(A x B) x B"

It should be:

B x (A x B)

or:

(B x A) x B

(B x A) is indeed out of the page as the text says but (A x B) x B has opposite direction to A || B.

I've also tried to implement the "Alternative using Matrix representation" without success. Since I'm pressed for time, I didn't try to find out why.