I know that if the dot product of every combination equals zero, the set is orthogonal. No problems here. I do that, and they all equal zero.

a dot b:

1 * 0 + -2 * 1 + 1 * 2 = 0

a dot c:

1 * -5 + -2 * -2 + 1 * 1 = 0

b dot c:

0 * -5 + 1 * -2 + 2 * 1 = 0

The problem comes with another theorem I learned to determine if a set is orthogonal. I thought if i made a square matrix out of orthogonal sets, the matrix is also orthogonal. Thus, the matrix's transpose equals its inverse and its transpose times the original equals an identity matrix.

So I check what I've already concluded this time with the matrix method:

[ 1, -2, 1]T [1, -2, 1] --------------------[1,0,0]

[ 0, 1, 2] * [ 0, 1, 2] which doesn't equal [0,1,0]

[-5, -2, 1] [-5, -2, 1] -------------------- [0,0,1]

I tried making the vectors columns and rows(from my understanding, both should work), and there is an example in my notes that uses this method. The example works. this does not. It's really frustrating.

I'm using a calculator to do the matrix math.