p=orthogonal which means:
Orthogonalize each column, and solve above identity to find a,b,c making p:
Use the Gram-Schmidt process to find last row of the following orthogonal matrix where the first two rows are:
[sqrt(3)/2, -1/sqrt(8), 1/sqrt(8)]
[0, 1/sqrt(3), sqrt(2)/sqrt(3)]
(third row is [a,b,c]).
Now, because the matrix is orthogonal, all of the rows are perpendicular and have length 1. To find the third row, call it q3, using Gram-Schmidt, we would do this-
q3 = [a,b,c] - proj q1 ([a,b,c]) - proj q2 ([a,b,c])
where q1 = [sqrt(3)/2, -1/sqrt(8), 1/sqrt(8)] and q2 = [0, 1/sqrt(3), sqrt(2)/sqrt(3)].
When I try and brute force this thing and solve, I get crazy fractions etc. I must be missing something...