Find two vectors v and w such that the three vectors u = (1,-1,-1), v and w are linearly independent independent.
Three vectors are linearly independent if the determinant formed by the vectors (in columns) is non-zero.
So for u=(1,-1,-1), v=(a,b,c), w=(d,e,f)
There are many possible choices of v and w such that the determinant
1 a d
-1 b e
-1 c f
is non-zero.
The simplest way is to create a triangular matrix such that the diagonal is all non-zero, or
1 0 0
-1 b 0
-1 c f
where b and f are non-zero, and the determinant evaluates to b*f≠0.
Example: (b=-1,c=1,f=1)
(1,-1,-1),(0,-1,1),(0,0,1) are linearly independent because the determinant
1 0 0
-1 -1 0
-1 1 1
evaluates to 1*(-1)*(-1)=-1 ≠ 0
Note:
vectors that are orthogonal to each other are linearly independent, since each cannot be a linear combination of the others.
However, linear independent vectors need not be orthogonal. Therefore their dot products need not be zero.