Unit vector n perpendicular

Hi!

I am having a problem with the following, can anyopne help?

Find a unit vector u perpendicular to both the vectors

p = (2,1,2) and q = (2,2,0)

Here's were I have got

Let u = (x,y,z)

|u| = 1 -> |u|^2 = 1

-> x^2+y^2+z^2 = 1 (1)

u . p = 0 -> x*2 + y*1 + z*2 = 0

-> 2x +y +2z = 0 (2)

u . q = 0 -> x*2 +y*2 + z = 0

-> 2x + 2y =0

-> x = -y (3)

Substitute z for y in (1) and (2)

x^2+z^2+z^2 = 1 -> x^2 + 2z^2 = 1 (1')

2x + z + 2z = 0 -> 2x + 3z = 0 (3')

Substitute -2x for z in (1')

x^2 + 2(-2x)^2 = 1 -> 9x^2 =1 -> x = +/- 1/3

When x = 1/3 y = -1/3 and z = -2/9 using (3) and (3')

but (1/3)^2 + (-1/3)^2 + (-2/9)^2 does not equal 1 as it should according to (1)

I am not sure what to do next. Can anyone help?