If you take the cross product of the two vectors, you will get a vector that is orthogonal to both of the vectors, and as such, is also orthogonal to the plane they lie in.

So

.

Now, if you were to take the dot product of this new vector with either or , you should get 0.

You need a UNIT vector in this direction. So divide it by its length.

.

So the unit vector is

.