How do you find all unit vectors orthogonal to v=i+j+k?

A generalised unit vector is:
##mathbf u = 1/(sqrt( 2 (alpha_2 ^2 + alpha_3 ^2 + alpha_2 alpha_3)))((- alpha_2 – alpha_3),(alpha_2),(alpha_3))##
There are an infinite number of vectors thare are orthogonal to ##mathbf v = ((1),(1),(1))##.
If ##mathbf alpha## is one such vector, we know from that ##mathbf v * mathbf alpha = 0 implies alpha_1 + alpha_2 + alpha_3 = 0##
A generalised vector is therefore:
##mathbf alpha = ((- alpha_2 – alpha_3),(alpha_2),(alpha_3))##
A generalised unit vector is:
##mathbf u = 1/(sqrt( 2 (alpha_2 ^2 + alpha_3 ^2 + alpha_2 alpha_3)))((- alpha_2 – alpha_3),(alpha_2),(alpha_3))##