Given the vectors <1/2, 1/2, 1/2, 1/2>,<1/2, 1/2, -1/2, -1/2>,<1/2, -1/2, 1/2, -1/2>, I'm asked to find a vector such that all four will be orthonormal. On such vector is <1/2, -1/2, -1/2, 1/2>. I'm also asked to find how many such vectors there are (given that the vector has to be a unit vector). Intuition from looking at the first two dimensions tells me that the answer is likely two; however I don't know how to prove this.