yup, I know that (U dot V) is = 1 because U dot V = (u1 * v1 + u2 * v2 +...+uN * vN)
I think it see the problem more clearly now: Is it that we create a vector x and dot multiply it by both vectors U and V and then pick values that will give the dot product between X and V and X and V = 0 for both cases?
In this case x_1 =x_2 = -x_3 works to make both dot products zero so we can create X = (1, 1, -1).
If this is the correct procedure, is there a general way of creating an orthogonal vector, or is it done by inspection?