I know I didn't post this question but I haven't got a clue how to do it either.
After reading your post, here's what I tried:
Therefore I just need to arrange and in the matrix so that each vector is linearly independent from the other ones. I got this:
?
is fine for the middle column, but the other two columns are not orthogonal to it. You could for example use for one of them. Can you see a vector (hint: without any fractions in its entries) that is orthogonal to both of these?
Just to emphasise the point: linear independence is not enough, the columns must be orthogonal to each other.