a question about column space

Let us assume there are three vectors x1, x2, x3 in R^n and they are
linearly independent.

First, let us choose x1 and generate two new vectors
x3-x1, x2-x1
Let y1 denote the orthonormal basis of the column space of x3-x1, x2-
x1

Similarly, we can generate
x3-x2, x1-x2
Let y2 denote the orthonormal basis of the column space of x3-x2, x1-
x2

x2-x3, x1-x3
Let y3 denote the orthonormal basis of the column space of x2-x3, x1-
x3

It seems that y1*y1' = y2*y2' = y3*y3'. Here y1' is the matrix
transpose of y1.

I was wondering whether we can prove the above observation to be true
or not?

Thanks
Roy

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