Your Gram-Schmidt process is a little off.
You should have: b2=normalize( a2-(a2.b1)b1 )
Since a 3rd vector a3 is not specified, you can use any vector as long as it is not a linear combination of a1 and a2.
That is, it must point out of the plane defined by a1 and a2 into the 3rd dimension.
Both the vectors you mention satisfy that requirement.