To find the vector that is in the same plane as the points c, a, and b, which is orthogonal to a we must:
1) first find the vector that is orthogonal to the plane CAB that passes through point b.
2) Rotate a about this vector 90 degrees.
To do this we can use the procedure described by @Bjorn Gustavsson(vector g is the plane that is orthogonal to plane c,a, and b)
Then, using the rodrigues equation (Rodrigues' rotation formula - Wikipedia), we plug in a for v, and k for g with theta being pi/2 and finally we obtain our correct vector.