Find all the unit vectors in the plane determined by **u**=(3,0,1) and **v**=(1,-1,1) that are perpendicular to the vector **w**=(1,2,0).

Why can't I solve this using the following method (and what is the best method using the cross product)?

$ \mathbf{y} = a.\mathbf{u} + b.\mathbf{v} \qquad (a,b\in \Bbb{R}) $

**y** gives me any point on the plane.

$ c.\mathbf{w} = \mathbf{v} \times \mathbf{y} \qquad (c\in \Bbb{R}) $

Cross product gives me the perpendicular vector.

Find **y** such that a,b solve the above equation then divide by ||**y**|| to get $ \mathbf{\hat y} $. I have tried it 3 times and each time I arrive at the nonsense answer.