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.
dwsmkth, the three given vectors are already orthogonal so "Gram-Schmidt" is not necessary.
Any vector, <a, b, c, d> that is normal to the three given must satisfy (1/2)a+ (1/2)b+ (1/2)c+ (1/2)d= 0, (1/2)a+ (1/2)b-(1/2)c-(1/2)d= 0, and (1/2)a-(1/2)b+ (1/2)c- (1/2)d= 0. An obvious first step is to multiply all equations by 2 to get a+ b+ c+ d= 0, a+ b- c- d= 0, and a- b+ c- d= 0. Adding the first two equations gives 2a+ 2b= 0 so b= -a. Adding the last two gives 2a-2d= 0 so d= a. Putting b= -a and d= a into the third equation, a- a- c- a= -c-a= 0 so c= -a. That is, any vector perpendicular to all of the three given vectors is of the form <a, -a, -a, a>. The length of that vector is. Note the "absolute value"-
since a itself can be either positive or negative while the square root is always positive. There are exactly two values of a that make 2|a|= 1.
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Originally Posted by Lord Voldemort 
