Find a vector of length 1 which is perpendicular to both u and v. Do not use cross product. u = (-3, 1, 2). v = (4, 0, -8)

How do start with the question? My lecturer haven't discuss cross product with us yet so I think that's why he specified to not use cross product.

So, I tried using dot product to find the equation of plane by letting n = (a, b, c) but I currently have no idea of how to find the variables. Have I missed out something? Or am I using the wrong method?

It's solving simultaneous linear equations. You've got the unknown vector x satisfying 2 equations:

u.x = 0
v.x = 0

and finally the normalisation condition gives you the final equation:

x.x = 1

ty.